CHARGE COLLECTION MECHANISMS IN A SUB-MICRON GRATED MSM PHOTODETECTOR: FIELD ANALYSIS

Transcription

1 CHARGE COLLECTION MECHANISMS IN A SUB-MICRON GRATED MSM PHOTODETECTOR: FIELD ANALYSIS A Thess pesented to the faculty of the Gaduate School Unvesty of Mssou-Columba In Patal Fulfllment Of the Requements fo the Degee Maste of Scence by SURESH KRANTHI NAKKA D. Naz E. Islam, Thess Supevso JULY 5

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3 ACKNOWLEDGEMENTS I would lke to acknowledge the help of many people dung my couse of study. Fstly, I would lke to thank my advso D. Naz E. Islam fo povdng nvaluable nsghts, tmely encouagement as well as gudance, balanced by the feedom to expess myself thoughout ths eseach wok. I am thankful to D. Robet Mclaen and D. Nel Fox who eadly ageed to be membes of my thess examnng commttee. I would lke to extend my sncee thanks to D. Phumn Kawanch fo many nsghtful convesatons and helpful comments whch made ths thess possble. In the ace aganst the clock n whch one almost nevtably becomes enmeshed when pepang a thess, t s a tue elef to be suounded by people who show sympathy and who contnue to beleve n what you ae dong. The cedt fo such suppot goes to my fends and colleagues, whose nspng nteest and ceaseless enthusasm fo the topc exploed n ths thess has meant a geat deal to me. Fnally, I dedcate ths wok to my paents and sstes. Wthout the love, undestandng, endless patence, suppot and encouagement ths wok could not have been accomplshed.

4 TABLE OF CONTENTS ACKNOWLEDGEMENTS. LIST OF ILLUSTRATIONS...v LIST OF SYMBOLS.v CHAPTER. INTRODUCTION.... BACKGROUND.5. Photodetecto. 5.. Pefomance Cteon fo a Good Photodetecto..6. Metal Semconducto Metal (MSM) Photodetecto..6.3 Electomagnetc Wave Fundamentals...3. Electomagnetc Waves Wave Equaton..3.3 Popagaton of Electomagnetc Waves Reflecton and Tansmsson of an Electomagnetc Wave.7.4. Nomal Incdence on a Lossless Delectc Oblque Incdence at a Delectc Bounday Pependcula Polazaton.....

5 .4.. Paallel Polazaton Enegy Tanspoted by the EM Waves LITERATURE REVIEW THE SIMULATION METHOD Intoducton The Fnte Integaton Technque (FIT) Implementaton of FIT Equatons nto Smulaton Softwae Tansent Solve Egenmode Solve Modal Analyss Solve Fequency Doman Solve RESEARCH APPROACH Intoducton Analyss of Squae and Wall lke Gatng Stuctues Analyss of Squae Gatngs wth Change n Aea Analyss of Cone lke Gatng Stuctues Analyss of Squae Gatngs wth Claddng CONCLUSIONS REFERENCES..7 v

6 LIST OF ILLUSTRATIONS Fgue Page. Geneal Stuctue of a MSM Photodetecto Basc Schematc of an Intedgtated MSM Photodetecto Topology of the Electomagnetc Waves..4 Spatal Vaatons of E and H Felds fo an EM Wave tavelng n z- decton Unfom plane wave nomally ncdent on a lossless delectc bounday A pependculaly polazed wave ncdent on a delectc bounday A paallel polazed wave at a delectc bounday Allocaton of the gd voltages and facet fluxes on the pmay and dual gds Tme devatve of the magnetc flux defned on the enclosed pmay cell facet Smulated stuctue wthout any gatngs Input exctaton sgnal (contnuous sne wave) Incdent wave ampltude changes as t tavels though the suface of the detecto wthout any gatngs Smulated Stuctue fo (a) walls and, (b) squae Gatngs Incdent wave ampltude changes as t tavels fom the wall-lke gatng to the substate egons Incdent wave ampltude changes as t tavels fom the squae gatng to the substate egons..54 v

7 5.7 Sngle wall gatng on the actve egon used fo analyss Font Vew of the Smulated Stuctue wth dffeent S extenson aeas Incdent wave ampltude changes fo a wave tavelng though dffeent S extenson aeas Plot of the E-feld vs numbe of spkes Smulated stuctue wth cone gatngs Incdent wave ampltude changes as t tavels though cone and squae gatngs Analyss of (a) cone and, (b) squae gatng stuctues Smulated Stuctue wth dopng aound the squae gatngs Incdent wave ampltude changes as t tavels though squae gatngs wth and wthout dopng (coatng) Gudng of waves n a squae gatng.66 v

8 LIST OF SYMBOLS Symbol Physcal Quantty Unt ψ Electostatc Potental Volt, V q Electon chage Coulomb, C c Speed of lght Metes pe sec, m/s k Popagaton constant/wave numbe Radans pe mete, ad/m λ Wavelength Mete, m f Fequency Hetz, Hz E Electc feld Intensty Volts pe mete, V/m H Magnetc Feld Intensty Ampee pe mete, A/m D Electc Flux Densty Coulomb pe squae mete, C/m B Magnetc Flux Densty Tesla, T µ Pemeablty Heny pe mete, H/m ε Pemttvty Faad pe mete, F/m J c Cuent Densty Ampee pe squae mete, A/m ρ v Volume chage densty Coulomb pe cubc mete, C/m 3 S Poyntng Vecto Watts pe squae mete, W/m S av Powe Densty Watts pe squae mete, W/m σ Conductvty Semens pe mete, S/m v

9 η Intnsc Impedance Ohm, Ω ω Angula Fequency Radans pe second, ad/s n Refactve Index Γ Reflecton Coeffcent τ Tansmsson Coeffcent Z Chaactestc Impedance Ohm, Ω θ Angle of Incdence Electcal Degees, o θ Angle of Reflecton Electcal Degees, o θ t Angle of Tansmsson Electcal Degees, o v

10 CHAPTER INTRODUCTION A Photodetecto s a semconducto devce that absobs optcal enegy and convets t to electcal enegy, whch usually manfests as a photocuent. It s a key component n optcal tansmsson and measuement systems. Photodetectos ae mpotant n optcalfbe communcaton systems n the nea-nfaed egon. They demodulate optcal sgnals, that s, convet the optcal vaatons nto electcal vaatons, whch ae subsequently amplfed and futhe pocessed. Fo such applcatons, photodetectos must satsfy stngent equements such as hgh senstvty at opeatng wavelengths, hgh esponse speed, and maxmum effcency. Hgh effcency and low-powe photodetectos (PD s) ae always sought afte n both long-haul and local aea communcaton systems. Hgh speed PD s ae used n a wde ange of mcowave photoncs applcatons fom fbe optc communcaton lnes and optcal weless systems to photonc measuement systems fo detecton and conveson of optcal sgnals, and fo mcowave geneaton, as well as fo optcal contol of mcowave ccuts and devces. Photodetectos ae a ctcal component n lght wave systems. As data ates ncease, mplementng hgh-quantum effcent, shot esponse tme, low capactance (fo nput to eceves), and lage-aea (fo senstvty and algnment toleance) photodetectos

11 becomes a ctcal ssue. Howeve, hgh effcency, shot esponse tme, and a lage detecton aea nvolve tadeoffs that must be ntellgently optmzed. A specal knd of photodetecto called the Metal Semconducto Metal (MSM) photodetecto s attactve fo many optoelectonc applcatons, such as optcal communcatons, futue hgh-speed chp-to-chp connecton, and hgh-speed samplng, because of the hgh senstvty-bandwdth poduct and the compatblty wth lagescale plana ntegated ccut (IC) technology. The electodes of the MSM photodetecto ae often ntedgtated to ncease the actve egon aea, whle optmzng the electc felds n the cae collecton egon. An ncease n the collecton effcency of S MSM photodetectos theefoe would make these devces moe attactve fo a wde vaety of applcatons. One pocess to ncease the collecton effcency of the MSM photodetectos s to use wall-lke gated stuctues of submcon dmensons on the photodetecto. Ths concept s used as a benchmak fo ths poject. An ncease n chage collecton effcency of a MSM Photodetecto can be analyzed though two dffeent methods: one s the semconducto appoach and othe s the electomagnetc appoach. In ths poject, the electomagnetc appoach was chosen as the smulaton tool.

12 The man objectve of ths poject s to pefom electomagnetc feld analyss to explan the mpoved collecton effcency of gated photodetectos. Ths analyss of a S MSM photodetecto s pefomed by lookng nto the tansmsson of electomagnetc wave feld components as they tavel though the gated stuctues on the detecto nto the devce s actve egon. Hee, t s shown that thee s an ncease n the collecton effcency fo one gven wavelength of the ncdent electomagnetc sgnal when the walllke gatng stuctues on the detecto ae changed to a squae lattce. The vaatons n the ampltude of the electc feld ntensty of the photodetecto wee detemned by changng the aea coveed by the S gatngs on the detecto. All the smulatons wee done usng the CST Mcowave Studo softwae whch s based on the Fnte Integaton Technque. Ths thess s compsed of sx chaptes whch ae oganzed as follows. Chapte povdes the backgound of a photodetecto and ts bascs. It explans the pefomance ctea of a good photodetecto. Ths s followed by a dscusson of a specal knd of detecto called an MSM Photodetecto and ts functonng. The fundamentals of electomagnetc waves and the enegy tanspoted by these waves ae also pesented. Fnally, the eflecton and tansmsson of the EM waves between two meda togethe wth equatons elated to the effcency of the EM waves fo the case of nomal and oblque ncdence ae pesented n detal. 3

13 Chapte 3 summazes pevous eseach wok done on the MSM photodetectos. It talks about the vaous attempts nvolved to ncease the effcency of an MSM photodetecto. Chapte 4 stats wth the ntoducton of dffeent numecal technques used to solve the complcated electomagnetc poblems, followed by a bef ntoducton of the Fnte Integaton Technque (FIT). Ths chapte ends wth the ncopoaton of the appopate equatons nto the softwae used fo the analyss. Chapte 5 deals wth the smulaton setup and the esults obtaned. The eseach wok done to ncease the chage collecton effcency of the MSM photodetecto s also dscussed. It s shown that the squae-lke gatngs mpove the effcency of the photodetecto, as compaed to a wall-lke stuctue n the actve egon. Smulaton esults along wth explanatons fo the case of vaatons n the numbe of S gatngs and vaatons n ts dmensons ae also pesented. Fnally, n Chapte 6, conclusons ae dawn and some deas fo futhe study ae dscussed. 4

14 CHAPTER BACKGROUND. Photodetecto Photodetectos ae used n many applcatons of eveyday lfe, fom the ba code scanne at the gocey stoe to the eceve fo a emote contol on a VCR, as well as the photoeceve at the end of a fbe optc cable n an optcal communcaton system. A photodetecto s an optoelectonc devce that absobs optcal enegy and convets t to electcal enegy, whch usually manfests as a photocuent. Thee ae geneally thee steps nvolved n the photodetecton pocess: ) Absopton of optcal enegy and geneaton of caes ) Tanspotaton of photogeneated caes acoss the absopton and/o tanst egon 3) Cae collecton and geneaton of a photocuent, whch flows though extenal ccuty The pocess of photodetecton s sometmes assocated wth demodulaton, when a hghfequency optcal sgnal s conveted nto a tme-vayng electcal sgnal and futhe pocessed and ectfed. Photodetectos ae used to detect optcal sgnals angng ove a vey wde ange of the optcal spectum. The hgh data ate of the pesent day optcal fbe tansmsson systems mposes sevee demands on the esponse speed of the photodetecto. In ths applcaton, detectos should eceve the tansmtted optcal pulses 5

15 and convet them, wth as lttle loss as possble, nto electonc pulses that can be used by a telephone, a compute, o othe temnal at the ecevng end... Pefomance Cteon fo a Good Photodetecto A photodetecto must satsfy vey stngent equements fo effectve pefomance and compatblty. The man pefomance ctea fo good photodetectos ae:. Hgh senstvty at the opeatng wavelengths. Hgh fdelty 3. Lage optcal to electcal conveson effcency 4. Hgh esponse speed 5. Lage SNR at the output 6. Hgh elablty 7. Low senstvty of pefomance to ambent condtons Because the photodetecto s only a pat of the whole optoelectonc eceve system, most of whch s electonc n natue, the desgn of the detecto should be compatble wth the desgn and achtectue of the est of the system. Ths compatblty eques that the detecto should have a small-sze, a low-bas voltage, and can be easly ntegated nto the eceve system.. Metal Semconducto Metal (MSM) Photodetecto Metal-Semconducto-Metal (MSM) photodetectos ae a specal knd of photodetectos whch ae attactve fo many optoelectonc applcatons ncludng the next geneaton of 6

16 hgh pefomance optcal communcaton nteconnects. MSM s ae smple to fabcate and ae compatble wth VLSI technology. MSM PD s have smple devce technology, fast esponse, small capactance and a lage actve aea. Metal Semconducto Metal (MSM) photodetecto bascally conssts of two Schottky baes connected back to back as shown n Fg... Lght s eceved at the gap between the metal contacts, and the MSM photodetecto avods absopton of lght by the metal laye as n a conventonal Schottky photodode. Fo compound semconductos, the lght absopton laye s usually deposted on a sem-nsulatng substate. Schottky contacts Undoped actve laye Sem-Insulatng Substate Fg.. Geneal Stuctue of a MSM Photodetecto Metal-semconducto-metal (MSM) photodetectos offe an attactve beneft ove altenatve photodetectos, such as conventonal p--n photododes. An MSM photodetecto s nheently plana and eques only a sngle photolthogaphy step, whch s compatble wth exstng feld effect tanssto (FET) technology. MSM photodetectos ae vey hgh speed devces due to the low capactance, and they typcally have vey low dak cuents (cuent poduced wthout ncdent lght). Howeve, the esponsvty 7

17 (total sgnal poduced fom a gven optcal nput) s qute low compaed to p--n photododes. The man causes fo the low esponsvty s the eflecton fom the metal suface and semconducto suface, the fnte cae lfetme as the caes tavese the gap between the electodes befoe beng collected, absopton of ncdent lght outsde the egon n whch photogeneated caes can be collected by the electodes, and suface ecombnaton cuents and deep taps wthn the semconducto mateal, whch may lowe the detected optcal sgnal. MSM photododes have a much lowe capactance pe unt aea than p--n photododes and thus ae often tanst-tme lmted. The tanst tme s elated to the spacng between ntedgtated electodes. A MSM photodode can be used to mpove the feasblty of fabcatng optoelectonc ntegated ccuts (OEIC s) fo a fbe optc communcaton system because of ts smple and compatble fabcaton pocess. The basc stuctue of an ntedgtated MSM photodetecto s shown n Fgue.. Metal Contact Ulta Thn Metal Box Oxde Semconducto Fg. Basc Schematc of an Intedgtated MSM Photodetecto 8

18 The ntegablty of MSM photododes wth pe-amplfe ccuty comes fom the fact that. MSM photododes do not eque dopng, whch elmnates any paastc capactve couplng between the photodode and the doped egons wthn the actve tansstos; and. The Scottky electodes of the MSM photododes ae essentally dentcal to gate metallzaton of feld effect tansstos (FET s). MSM photodetectos suffe fom vey low quantum effcences also because the metallzaton fo the electodes shadows the actve lght collectng egon. Shadowng can lmt the ncdent lght fom eachng the actve egon of the MSM detecto and pevents the quantum effcency fom beng moe than 5% fo equal electode wdths and spacng. Thee ae desgn tade-offs n MSM photododes fo optmzng the speed and quantum effcency. The aveage cae tanst tme n an MSM photodode can be deceased by educng the absopton laye thckness, nceasng the appled bas, o educng the ntedgtated electode spacngs. Howeve, a decease n the absopton laye thckness esults n the degadaton of esponsvty; a decease n the electode spacng leads to a degadaton of the dak cuent and the beakdown voltage and the equement fo complex lthogaphy. An analyss of the enegy tanspoted n a MSM Photodetecto s possble though two dffeent methods. One s the semconducto appoach and othe s the electomagnetc appoach. 9

19 In the semconducto appoach the thee basc equatons that povde the geneal famewok fo the chage tanspot ae Posson s Equaton: dv ( ε ψ ) = ρ (.) Cae Contnuty Equatons n = dvj t q n + G n R n (.) p = dvj t q p + G p R p (.3) Whee, ψ = Electostatc potental, ρ = Space chage densty, q = Magntude of the chage on an electon, n and p = electon and hole concentatons espectvely, J n and J p = electon and hole cuent denstes, G n and R n and G p R p = Geneaton ate of electon and holes, = Recombnaton ates fo electons and holes, In addton, seconday equatons ae used to specfy patcula physcal models fo electon and hole geneaton, ecombnaton ates and cuent denstes. The analyss of the enegy tanspoted n a MSM photodetecto can also be done usng electomagnetc appoach. In ths poject, the electomagnetc appoach s chosen as the smulaton tool.

20 .3 Electomagnetc Wave Fundamentals.3. Electomagnetc Waves James C. Maxwell (83-879) publshed hs Electomagnetc Feld Equatons n 864. He bought togethe pevous expemental wok and concepts of Gauss, Ampee and Faaday, as well as hs own knowledge of mathematcs to pesent ths analyss of electomagnetc felds. Wth hs feld equatons, Maxwell calculated the speed of electomagnetc popagaton to be the same as the speed of lght, whch ndcated that lght s also an electomagnetc feld. Fg..3 shows the basc topology of Electomagnetc waves. Maxwell s equatons ae the bass fo the theoy of electomagnetc felds and waves. They ae used n the desgn of antennas, tansmsson lnes, cavty esonatos, fbe optcs and solvng adaton poblems. ae descbed by a Electomagnetc Waves ae pat of Wave Equaton Povde Electomagnetc Spectum asng fom Enegy Tanspot and all tavel at Maxwell s Equatons d escbed by the Speed of Lght Poyntng Vecto Fg.3 Topology of the Electomagnetc Waves

21 .3. Wave Equaton The Maxwell s equatons n dffeental fom ae gven below H E = µ t (.4) E H = Jc + ε t (.5) D = ρ v (.6) B = (.7) Whee, E s Electc Feld ntensty (V/m), H s the magnetc feld ntensty (A/m), D s the Electc Flux Densty (C/m ), B s the Magnetc Flux Densty (T), µ s the pemeablty (H/m), ε s the pemttvty (F/m), ρ v s the Volume chage densty (C/m 3 ). J c s the cuent densty (A/m ), and In fee space o a lossless delectc, Eqs..5 and.6 become E H = ε (.8) t D = E = (.9) Then, by takng the cul of Equaton.4 and substtutng nto Eq..8, H E ( E) = µ = µε (.) t t But, ( E) = (. E) E (.) E t ( E) = (. E) E = µε (.)

22 Hence, E t E = µε V/m 3 (.3) When expanded n Catesan coodnates, Eq..3 becomes, E x E E + + y z E = µε V/m 3 (.4) t The above equaton s called Maxwell s EM Wave equaton. Thee ae many types of possble EM waves. All these possble EM waves must obey ths specal wave equaton that descbes the tme and space dependence of the electc feld. In an sotopc and lnea delectc medum, the elatve pemttvty s the same n all dectons and s ndependent of the electc feld. The electc feld and the magnetc feld ae mpotant concepts that can be used to mathematcally descbe the physcal natue of electomagnetc waves such as lght. The electc feld vbates tansvese (.e. pependcula) to the decton the electomagnetc wave s tavelng. The magnetc feld vbates n a decton tansvese to the decton n whch the electomagnetc wave s tavelng and tansvese to the electc feld. Fg..4 llustates the behavo of an electomagnetc wave that s polazed along the x- axs and tavelng n the z-decton. These two felds oscllate n a consstent manne so that the wave moves fowad at a constant ate, the speed of lght (c). Lght s an 3

23 electomagnetc wave wth tme vayng electc and magnetc felds, E x and B y espectvely, whch ae popagatng n space n such a way that they ae always pependcula to each othe. The decton of popagaton of the wave s n the z-decton. Electomagnetc (EM) waves ae poduced by movng chages. These ae changng electc and magnetc felds, cayng enegy though space. EM waves eque no medum; they can tavel though empty space. Snusodal plane waves ae one type of electomagnetc waves. Not all EM waves ae snusodal plane waves, but all EM waves can be vewed as a lnea supeposton of snusodal plane waves tavelng n abtay dectons. A plane EM wave tavelng n the z-decton can be descbed by, E x (z, t) = E o cosπ (ft z/ λ) (.5) E x Electc Feld Vaaton λ Z B y Magnetc Feld Vaaton Fg..4 Spatal Vaatons of E and H Felds fo an EM wave tavelng n z-decton 4

24 Whee, E x s the electc feld at poston z at tme t, k s the popagaton constant o wave numbe, whch s equvalent to (π/λ), λ s the wavelength, f s the fequency, and E o s the ampltude of the wave. The tme vayng magnetc felds esult n tme vayng electc felds and vce vesa. A tme vayng magnetc feld would set up a tme vayng electc feld wth the same fequency..3.3 Popagaton of Electomagnetc Waves Electomagnetc waves ae the waves whch can tavel though the vacuum of oute space. Mechancal waves, unlke electomagnetc waves, eque the pesence of a mateal medum n ode to tanspot the enegy fom one locaton to anothe. Sound waves ae examples of mechancal waves, whle lght waves ae examples of electomagnetc waves. Electomagnetc waves ae ceated by the vbaton of an electc chage. Ths vbaton ceates a wave whch has both an electc and a magnetc component. The popagaton of an electomagnetc wave though a mateal medum occus at a net speed whch s less than 3 x 8 m/s. The mechansm of enegy tanspot though a medum nvolves the absopton and eemsson of the wave enegy by the atoms of the mateal. When an electomagnetc wave mpnges upon the atoms of a mateal, the enegy of that wave s absobed. The absopton of enegy causes the electons wthn the atoms to undego vbatons. Afte a shot peod of vbatonal moton, the vbatng electons ceate a new electomagnetc wave wth the same fequency as the fst electomagnetc wave. Whle these vbatons occu fo only a vey shot tme, they delay the moton of the wave though the medum. 5

25 Once the enegy of the electomagnetc wave s e-emtted by an atom, t tavels though a small egon of space between atoms. Once t eaches the next atom, the electomagnetc wave s absobed, tansfomed nto electon vbatons and then eemtted as an electomagnetc wave. Whle the electomagnetc wave wll tavel at a speed of c (3 x 8 m/s) though the vacuum of nteatomc space, the absopton and e-emsson pocess causes the net speed of the electomagnetc wave to be less than c. The actual speed of an electomagnetc wave though a mateal medum s dependent upon the optcal densty of that medum. When a lght passes though a medum, ts velocty deceases. Fo a gven fequency of lght, the wavelength also must decease. Ths decease n velocty s quanttated by the efactve ndex, n, of the medum whch s the ato of c to the velocty of lght n that medum (v), n = c / v. Snce the velocty of lght s less n meda othe than n a vacuum, n s always a numbe geate than one. Dffeent mateals cause dffeent amounts of delay due to the absopton and eemsson pocess. Futhemoe, dffeent mateals have the atoms moe closely packed, and thus the dstance between atoms s less. These two factos ae dependent upon the natue of the mateal though whch the electomagnetc wave s tavelng. As a esult, the speed of an electomagnetc wave s dependent upon the mateal though whch t s tavelng. 6

26 .4 Reflecton and Tansmsson of an Electomagnetc Wave.4. Nomal Incdence on a Lossless Delectc When a unfom plane wave popagatng n medum s nomally ncdent on an nteface wth a second medum wth a dffeent delectc constant as shown n Fg..5, some of the ncdent wave enegy s tansmtted nto medum and contnues to popagate to the ght (+z decton). In the followng dscusson, t s assumed that both meda ae lossless delectcs (.e., σ σ ). Once agan, t s assumed that wthout loss, = of genealty unde condtons of nomal ncdence on a plana bounday, that the ncdent electc feld s oented n the x-decton. It s also assumed that the ampltude of the ncdent wave s eal, wth no loss of genealty, snce ths bascally amounts to the choce of the tme ogn []. The phaso felds fo the ncdent, eflected, and tansmtted waves ae gven as: E X Reflected Wave E E t Tansmtted Wave K H H t K t Incdent Wave E Y Z H Medum ε, µ η ) (, K Medum ε, µ η ) (, Z= Fg.5 Unfom plane wave nomally ncdent on a lossless delectc bounday 7

27 Incdent Wave: ˆ jkz E ( z) = xee (.6) E = η H ( z) yˆ jkz e Reflected Wave: ( ) = ˆ + jkz E z xee (.7) E η H z yˆ + jkz e ( ) = Tansmtted wave ( ) = ˆ t t jk z E z xee (.8) E ( ) = η t t H z yˆ jk z e Whee, k = ω µ ε, k = ω µ ε and η = µ /ε, η = µ /ε ae, espectvely, the wave numbe and the ntnsc mpedance fo medum and medum, espectvely. t Note that s the ampltude (yet to be detemned) of the tansmtted wave at z =. E Fom Fg..5 the polates of E and E have been defned to be the same and H to be n the y decton, so that E H s n the z decton. Note that, at ths pont, the selected oentatons of E and H fo the dffeent waves (ncdent, eflected and tansmtted) ae smply convenent choces. The bounday condtons wll detemne whethe the phaso felds at the bounday ae postve o negatve accodng to these assumed conventons. 8

28 Assumng the ncdent wave as gven, the next step s to detemne the popetes of the eflected and tansmtted waves so that the fundamental bounday condtons fo electomagnetc felds ae satsfed at the nteface, whee all the thee waves ae elated t to one anothe. Thee ae two unknown quanttes and to be detemned n tems of the ncdent feld ampltude E. Two bounday condtons wll be used to detemne them. The bounday condtons to be employed ae ) the tangental components of the electc feld should be contnuous acoss the juncton and ) the tangental components of the magnetc feld ntensty dffe by any suface cuent that s located at the nteface. It s easonable n pactce to assume that ths cuent s equal to zeo. Ths mples that the tangental components of the magnetc feld ntensty wll also be contnuous at the nteface. Thus, the two bounday condtons ae the contnuty of the tangental components of both the electc and magnetc felds acoss the nteface. We thus have, E E ( = ) + ( = ) = ( = ) + = t t E z E z E z E E E t t E E E H ( z = ) + H ( z = ) = H ( z = ) = η η η The soluton of these two equatons yelds, η η = (.9) E E η + η η = (.) t E E η + η The eflecton and tansmsson coeffcents ae defned as follows 9

29 Γ = E E η η = η + η (.) τ E E t = = η η + η (.) The quanttes Γ and τ ae called the Reflecton Coeffcent and Tansmsson Coeffcent, espectvely. Fo lossless delectc meda, η and η ae eal quanttes; consequently, both Γ and τ ae eal also. Note that, physcally, the above coeffcents ae deved fom the applcaton of the bounday condtons, whch ae vald fo all meda n geneal. Complex eflecton and tansmsson coeffcents may esult when η and/o η ae complex (.e., one o both of the meda ae lossy), meanng that n addton to the dffeences n ampltudes, phase shfts ae also ntoduced between the ncdent, eflected and tansmtted felds at the nteface. Fom Eqs. (.) and (.) t can be easly shown that Γ and τ ae nteelated by the smple fomula, Fo nonmagnetc meda, τ = + Γ (.3) η = η = η ε, η ε, (.4) Whee, η s the ntnsc mpedance of fee space, n whch case Eqs. (.) and (.) may be wtten as,

30 ε Γ = (.5) ε + ε ε τ = (.6) ε ε + ε Fo most delectcs and nsulatos, the magnetc pemeablty does not dffe appecably fom the fee space value. Hence, µ = µ = µ and snce the chaactestc mpedance, Z = µ /ε. Eqs. (.5) and (.6) can be wtten as, Z Γ = (.7) Z + Z Z Z τ = (.8) Z + Z Hence, knowng the chaactestc mpedance of the mateals allows one to detemne the popagaton chaactestcs and ampltudes of the wave that s tansmtted nto the second mateal and of the wave that s eflected at the nteface and popagates back nto the fst mateal. If the chaactestc mpedances on both sdes of the nteface ae equal, all of the ncdent electomagnetc enegy wll be tansmtted nto egon and none eflected back nto egon. Ths s called matchng the meda, whch has many pactcal applcatons..4. Oblque Incdence at a Delectc Bounday Fo nomal ncdence, the eflecton coeffcent Γ and tansmsson coeffcent τ of a bounday between two dffeent meda s ndependent of the polazaton of the ncdent

31 wave, because the electc and magnetc felds of a nomally ncdent plane wave ae both always tangental to the bounday egadless of the wave polazaton. Ths s not the case fo oblque ncdence at an angle θ. A wave wth any specfed polazaton may be descbed as the supeposton of two othogonally polazed waves, one wth ts electc feld paallel to the plane of ncdence (paallel polazaton) and anothe wth ts electc feld pependcula to the plane of ncdence (pependcula polazaton) []. These two knds of polazatons wll now be befly dscussed..4.. Pependcula Polazaton The expessons fo the wave electc and magnetc feld phasos of the ncdent, eflected, and efacted (tansmtted) waves shown n Fg..6 can be expessed as Incdent Wave: E x, z) = ye ˆ e ( jk ( x snθ + z cosθ ) (.9) H E jk ( x snθ + z cosθ ) ( x, z) = ( xˆ cosθ + zˆsnθ) e η Reflected Wave: E x z = ye ˆ e jk ( x snθ z cosθ ) (, ) (.3) H E jk ( x snθ z cosθ ) ( x, z) = ( xˆ cosθ + zˆsnθ ) e η Tansmtted wave E x z = ye ˆ e t t jk ( x snθ t + z cosθt ) (, ) (.3)

32 H E t t jk ( x snθt + z cosθt ) ( x, z) = ( xcosθt + zˆsnθt ) e η X K H Tansmtted Wave E t K t E Reflected Wave θ θ Y θ t Z H t Incdent Wave E Medum ( ε µ, σ, = K H Z= Medum ( ε µ, σ ), = Fg..6 A pependculaly polazed wave ncdent on a delectc bounday To detemne the ampltudes of the eflected and tansmtted wave felds n tems of the ncdent feld ampltude E, one can apply the bounday condton concenng the contnuty of the tangental component of the wave electc feld acoss the nteface. Consdeng the feld oentatons as defned n Fg..6, then, at z =, E e θ + E e = E e jkx sn jkx snθ t jk x sn θ t (.3) Snce ths condton has to be satsfed fo all values of x, all thee components must be equal. Thus, k xsn = k x kxsnθ (.33) θ snθ = t The fst equalty n (.33) leads to 3

33 θ = θ (.34) Eq (.34) s commonly efeed to as Snell s law. The second equalty n (.33) leads to snθ snθ k k = = = t ω ω µ ε µ ε n n (.35) Eq. (.35) s commonly efeed to as Snell s law of efacton. Rewtng the bounday condton at any gven value of x, (say at x=) E E E + (.36) E t t E = E = + E On the bass of the consevaton of powe we must have, S av cos θ = S cosθ + S cosθ (.37) av av t t E cosθ E cosθ E = + η η η t cosθ t o t ηe cosθt = (.38) ηe cosθ E E Now substtutng Eq. (.36) nto Eq. (.38) and manpulatng the esult to solve fo E E (by elmnatng t E ), the esult s, E η cosθ η cosθ t Γ = = (.39) E η cosθ + η cosθt 4

34 Whee, Γ s called the eflecton coeffcent fo pependcula polazaton. Fo magnetcally dentcal meda ( µ = ), and usng Eq. (.35), one can obtan the altenate expesson fo Γ, µ E cosθ ε / ε cosθt cosθ ( ε / ε ) sn θ Γ = = = (.4) E cosθ + ε / ε cosθ cosθ + ( ε / ε ) sn θ t The tansmsson coeffcent τ can be found fom Eqs. (.38) and (.39) as τ E η cosθ t = = (.4) E η cosθ + η cosθt Fo magnetcally dentcal meda ( µ = µ ), Eq. (.4) becomes, τ cosθ cosθ = = (.4) cosθ + ε / ε cosθt cosθ + ε / ε sn θ Hence, + Γ = τ (.43).4.. Paallel Polazaton The expessons fo the wave electc and magnetc feld phasos of the ncdent, eflected, and efacted (tansmtted) waves shown n Fg..7 can be expessed as Incdent Wave: E ( x, z) = E ( xˆ cosθ z θ ) e ˆsn jk ( x snθ + z cosθ ) (.44) 5

35 H E = yˆ e η (, ) x z jk ( x snθ + z cosθ ) X K E E t Tansmtted Wave K t H Reflected Wave θ θ Y θ t H t Z Incdent Wave E H Medum ( ε µ, σ, = K Z= Medum ( ε µ, σ ), = Fg.7 A paallel polazed wave at a delectc bounday Reflected Wave: E ( x, z) = E ( xˆ cosθ + z θ ) e ˆsn jk ( x snθ z cosθ ) (.45) H E = yˆ e η (, ) x z jk ( x snθ z cosθ ) Tansmtted wave E t jk t t ( x, z) = E ( xˆ cosθ z θ ) e t t ˆsn t ( x snθ + z cosθ ) (.46) H t t E = yˆ e η jk t t (, ) x z ( x snθ + z cosθ ) Followng a pocedue smla to that used fo the pependcula polazaton case to fnd the ampltudes of the eflected and tansmtted waves n tems of E, apply the bounday 6

36 condton concenng the contnuty of the tangental component of the wave electc feld acoss the nteface. Theefoe, at z =, t ( E + E ) θ E cos t cos = o θ E E t E + cosθ = E (.47) cosθt Substtutng Eq. (.47) nto Eq. (.38) and manpulatng the esult to solve fo E / E (by elmnatng t E ), the esult s, E η cosθ + η cosθ t Γ = = (.48) E η cosθ + η cosθt Fo magnetcally dentcal meda ( µ = µ ), cosθ ε / ε cosθ cosθ + ( ε / ε ) ( ε / ε ) sn ( ε / ε ) ( ε / ε ) sn t Γ = = (.49) cosθt + ε / ε cosθ cosθ + θ θ whch s the eflecton coeffcent fo paallel polazaton, then elmnatng E we fnd, τ E η cosθ t = = (.5) E η cosθ + η cosθt Fo magnetcally dentcal meda ( µ = µ ), cosθ ε / ε cosθ τ = (.5) cosθ + ε ( ε / ε )sn θ t = / ε cosθ cosθ + ε / ε Fom Eqs. (.49) and (.5) t s noted that, 7

37 + Γ cosθt = τ ( ) cosθ These wave popagaton concepts can be appled fo the analyss of an MSM photodetecto as the EM wave ncdent on the gatngs of the photodetecto eaches the devce actve egon..5 Enegy Tanspoted by the EM Waves Electomagnetc waves tanspot enegy though space. In fee space, ths enegy s tanspoted by the wave wth speed c. The magntude of the enegy flux, S, s the amount of enegy that cosses a unt aea pependcula to the decton of popagaton of the wave pe unt tme. It s gven by S = E H ( µ ) (.5) / Snce, fo electomagnetc waves H = E / c, µ s a constant called the pemeablty of fee space, µ = 4π 7 H / m. The Poyntng vecto s the enegy flux vecto. Its decton s the decton of popagaton of the wave,.e. the decton n whch the enegy s tanspoted. S = / µ ) E H (.53) ( Enegy pe unt aea pe unt tme s powe pe unt aea. S epesents the powe pe unt aea n an electomagnetc wave. If an electomagnetc wave falls onto an aea A whee t s absobed, then the powe delveed to that aea s 8

38 P = S da = S A A (.54) Ths concept of the tansmsson of enegy can be appled fo an MSM photodetecto as the wave, afte popagatng n the S gatng egon, eaches the actve egon of the detecto. Eq. (.54) wll be used n late chaptes fo analyzng the vaatons n the ampltude of the electc feld ntensty of an MSM photodetecto wth and wthout gatngs. 9

39 CHAPTER 3 LITERATURE REVIEW Photodetectos ae semconducto devces that can detect optcal sgnals though electonc pocesses. The extenson of coheent and ncoheent lght souces nto the fanfaed egon on one hand, and the ultavolet egon on the othe, has nceased the need fo hgh speed, senstve photodetectos. Hgh speed and hgh-senstvty photodetectos have been studed extensvely n the past ten yeas [], owng to the applcaton n boad-band optcal communcaton netwok and optcal geneaton of hgh-powe mcowave/mllmete waves [3]. R. G. DeCoby et al. [4] developed technques to mpove the speed and effcency of the photodetectos. In ths pape they dscussed how photodetectos ae geneally desgned wth a balance of bandwdth, effcency, and powe-handlng consdeatons that ae to be used n telecommuncatons and optoelectoncs. The Metal-Semconducto-Metal photodetecto (MSM PD), whch s a specal knd of photodetecto, was poposed and demonstated by Sugeta et al. n 979 [5]. MSM PD s have excellent potental as hgh-pefomance components fo hgh-speed lght wave communcaton systems and optoelectonc ntegated ccuts due to the low capactance pe unt aea and the hgh speed. MSM PD s deseve specal attenton due to the hgh electcal bandwdth and ablty to geneate ultashot electcal pulses [6]. 3

40 Metal-slcon-metal claddng layes exhbt couplng and absopton chaactestcs whch make them useful as photodetectos fo ntegated optcal applcatons. In the past few yeas MSM PD s have become vey popula n the feld of optcal communcatons because of the numeous advantages [3], [7]. One of the most mpotant popetes of ths type of detecto s ts hgh esponse speed, detemned by the geomety of the stuctue and by the low capacty of the detecto. The othe most mpotant popety s the effcency of the MSM photodetecto. The basc am n futhe development of MSM PD s s to acheve an mpovement of these popetes. Thee s theefoe an nceasng nteest n the modelng of MSM PD s and the compute smulaton of the esponse. Although ths devce has a hgh bandwdth, t suffes fom elatvely low quantum effcency due to hgh metal eflecton loss. Both S and GaAs metal-semconducto-metal (MSM) photodetectos ae vable canddates fo monolthcally ntegated optcal eceves n fbe optc communcatons and as fee space nteconnects [8]. GaAs s moe attactve due to ts shot absopton length (~. µm at λ = 85 nm), makng t possble to combne lage bandwdth wth good esponsvty [9]. These detectos opeate at vey hgh fequences (UV and vsble spectal ange). S offes the potental of lowe cost; dect ntegaton wth VLSI optoelectonc ccuts wth good senstvty, but poo esponse tmes have been epoted []. 3

41 Slcon MSM photodetectos have been used snce 96. The use of these slcon photodetectos s contnually gowng n vaous felds of scence such as astonomy, medcne, mateal testng, etc. Moe ecently, wth the fast development of botechnology, the need fo a photodetecto that woks n the UV ange wth hgh esponsvty has become clea. M. Caa et al. [] demonstated that commecal slcon photodetectos have a hgh esponsvty n the UV ange (-4 nm). In the UV ange of nteest, the absopton length n slcon s appoxmately 5 8 A o. They have nvestgated the popetes of dffeent types of commecal slcon photodetectos ncludng pxel and mcostp devces wth dffeent bulk and suface composton fom dffeent vendos. Lu et al. [] have epoted mpoved esponse speed by fabcatng MSM photodetectos on S-on-nsulato (SOI) substates. The key featue n speed enhancement s the bued oxde laye that lmts the actve S thckness. By educng the S flm thckness to nm, a photo detecto bandwdth of 4 GHz at a 78-nm wavelength was acheved, howeve, at the cost of vey low quantum effcency. Lee et al. [3] have poposed a MSM detecto confguaton on a 5-µm thck S membane, whee the tappng of lght n a thn membane esults n mnmal educton n esponsvty, whle educng cae tanst tmes. Othe attempts to mpove the absopton of S by hydogenated amophous S (a-s:h) have focused on modfyng the long-ange stuctual symmety of cystallne S by elaxng the k-selecton ule fo optcal tanston [4]. 3

42 Ove the past few yeas nteest has gown n the ntedgtated MSM detecto due to ts ease of ntegaton wth IC pocesses, ts vey low capactance, and ts hgh speed. Usng ths type of detecto, optcal communcaton eceves have been fabcated on ntegated ccuts. Intedgtated fnge MSM detectos have been used wdely as hgh-speed photodetectos and have also been used as (OE) optoelectonc mxes to geneate adofequency sub-caes n fbe optc mcowave lnks [5]. Recently, an MSM detecto has been utlzed as an OE mxe n a fequency modulated contnuous-wave lase detecton and angng (LADAR) system [6]. The esponse speed of the MSM-PD s lagely lmted by the tanst tme of the photogeneated caes, and thus the nte-electode spacng should be small. As the fabcaton technology advances, the fnge spacng of the MSM-PD deceases down to the sub-mcomete and even nanomete scale. MSM-PD s wth only 5 nm fnge wdth and spacng wee fabcated usng e-beam lthogaphy [7]. When the fnge spacng of a MSM photodetecto s smalle than the wavelength of lght, the tansmsson of TE and TM waves though the detecto fnges becomes stongly dependent on the wavelength and polazaton of the ncdent lght. Fo optmum pefomance of the MSM photodetecto, the amount of enegy eachng the nteface of the detecto should be maxmum, whch also depends on the geometc and optcal paametes of the stuctues as well as on the popetes of the ncdent adaton (wavelength, polazaton, angle of ncdence etc.). J.J. Kuta et al [8] demonstated how polazaton and wavelength account fo the esponse of a MSM photodetecto. Fo 33

43 stuctues whch ae lage compaed to the wavelength, suffcently good appoxmatons can be obtaned by means of smple geometcal optcs models. As featue szes become smalle, the eos caused by neglectng dffacton and ntefeence effects ncease. When the featue szes ae below the ode of a few wavelengths, goous electomagnetc models ae necessay to obtan easonably accuate esults. It was found fom [9] that appoxmately 3% of the ncdent lght s eflected at the nteface between the a and the detecto suface. If one neglects the gatng effect of fne metal fnges, these effects, n combnaton wth occultaton of the possble absobng suface aea by the metal electodes of, typcally, at least 5%, can educe the oveall quantum effcency to much less than 5%. A specal knd of detecto called the Resonant Cavty Enhanced (RCE) photodetecto, havng hgh quantum effcency was demonstated by Kshno et al. []. Ths s acheved by utlzng eflectos aound the actve egon. The photons make multple passes acoss the actve egon, mpovng the pobablty of absopton, theeby nceasng the quantum effcency. Some attempts have been amed at mpovng the S MSM detecto quantum effcency at vsble and nea IR wavelengths by fabcatng vetcal and U- shaped tench electodes usng eactve on etchng and wet chemcal etchng methods [], []. A. K. Shama [3] poposed a technque to mpove the effcency of the MSM Photodetecto. A smple on mplantaton step was used on a N-S-N metal- 34

44 semconducto-metal photodetecto to ceate a hghly absobng egon (~ µm) below the S suface, wheeby, the ntenal quantum effcency was mpoved by a facto of ~ 3 at 86 nm (up to 64 %) and a full facto of ten at.6 µm (up to 3 %) as compaed wth othewse dentcal non-mplanted devces. S. H. Zad and A. K. Shama [4] demonstated the pefomance of a S MSM photodetecto ncopoatng one-dmensonal (-D) aays of ectangula (wall) and tangula-shaped nanoscale stuctues wthn the actve egons. It has been shown that these gatngs account fo hghe tansmsson of enegy, theeby nceasng the chage collecton effcency. A new technque was poposed by Stéphane Colln et al. [5] fo effcent lght absopton n MSM photodetectos. It was shown that the confnement of lght n sub wavelength metal semconducto gatngs can be acheved by Faby Péot esonances nvolvng vetcal tansvese magnetc and tansvese electc guded waves, theeby nceasng the quantum effcency. Recently, Sang-Woo Seo et al. [6] demonstated a new knd of MSM photodetecto called an nveted metal semconducto metal (IMSM) photodetecto that has hghe effcency. These ae thn flm MSM s wthout the gowth substate. IMSM s have fnges at the bottom of the devce to enhance the effcency. 35

45 In ths thess, t s shown that havng a wall-lke S gatng on the actve egon of the detecto nceases the chage collecton effcency and eaangng the wall lattce to a squae lattce could mpove the collecton effcency futhe. It has been demonstated how these stuctual changes at the nteface accounts fo hghe tansmtted enegy and the subsequent geneaton and collecton of caes at the nteface, thus contbutng to enhanced collecton effcency. Vaatons n the electc feld ntensty due to the change n the numbe and the dmensons of squae gatngs ae also demonstated. It s also shown that claddng the detecto gatngs nceases the concentaton effcency of the photodetecto. 36

46 CHAPTER 4 THE SIMULATION METHOD 4. Intoducton The pevous chapte gave an dea about the pevous eseach wok done on MSM Photodetectos. It dscussed the vaous attempts to ncease the effcency of the photodetecto. Ths chapte outlnes the dffeent numecal technques used to solve the complcated electomagnetc poblems, followed by the basc concepts of the Fnte Integaton Technque (FIT). It concludes wth a bef ntoducton to the smulaton softwae used n the cuent analyss. Numecal technques ae extemely useful n solvng eal-lfe poblems wth complex mateals and geometes. Thee ae a vaety of electomagnetc modelng technques. Fo solvng complcated electomagnetc scatteng poblems, thee ae thee man numecal technques. Fst s the Fnte Dffeence Tme Doman (FDTD) technque, whch was fst poposed by Yee n 966 [7]. Ths technque s a computatonal method that calculates the tempoal evoluton of the electomagnetc feld wthn a egon of space by steppng though tme. At each tme step, centeed fnte dffeence appoxmatons ae used to calculate the space and tme dffeences on a Catesan gd. The electc and the magnetc felds ae defned by sx feld components, whch all le on a dffeent pont on the gd, especally defned to ft nto the FDTD scheme. Ths leads to an explct tme steppng algothm whch s of second ode accuacy n both tme and space. 37

47 The categoy of calculatons can be subdvded nto steady state and tansent analyss. In the steady state analyss, the exctaton s hamonc. Tme steppng s caed out untl steady state s eached,.e. all tansents have vanshed. The exctaton used fo the tansent analyss can have any tme functon. It may be an ncdent plane wave o an mpessed voltage o cuent at dscete ponts. The esult s a functon of tme, whch s tansfomed to the fequency doman wth a specal FFT to obtan the fequency esponse of the system. Local methods such as the FDTD method ae by necessty fomulated as ntal value poblems on a fnte poton of space (the computatonal doman). As such, the soluton of scatteng and adaton poblems eques a mechansm to enfoce the adaton condton whch pescbes the behavo of the electomagnetc felds at nfnty. Ths s acheved by applyng second-ode absobng bounday condtons whch tuncate the FDTD space gd and compensate fo eflectons on the bounday of the poblem space. The second method s the Electc Feld Integal Equaton (EFIE) method. The scatteng of abtaly-shaped pefectly-conductng bodes s solved usng a suface ntegal equaton fomulaton fo the electc feld. The electc feld (suface) ntegal equaton s solved by usng the method of moments n whch the testng functons of the electc feld and the expanson functons of the electc suface cuent densty ae both appopate tangula-patch functons. The system of equatons s then solved by usng a LU factozaton algothm. 38

48 EFIE method s applcable to both open and closed sufaces. Howeve, the EFIE fals nea ntenal esonances of a closed body. Seveal methods ae avalable fo elmnatng ths dffculty, but all nvolve sgnfcant addtonal computaton. Anothe dffculty s that the EFIE soluton pocedue becomes unstable when the dmensons of the scattee become vey small wth espect to the wavelength of the ncdent feld. The pncpal modfcaton to the ognal EFIE code ncludes the use of a new matx factozaton outne, whch esults n a sgnfcant educton n compute tme. The thd technque s the Fnte Integaton Technque (FIT). Ths technque s dscussed n detal n the followng secton, and ths method has been used fo all the smulatons n ths thess. 4. The Fnte Integaton Technque (FIT) The Fnte Integaton Technque (FIT), was fst poposed by Weland n 976/77 [8]. Ths numecal method povdes a unvesal spatal dscetzaton scheme, applcable to vaous electomagnetc poblems, angng fom statc feld calculatons to hgh fequency applcatons n the tme o fequency doman. In the followng secton, the man aspects of ths pocedue wll be explaned and aftewads extended to specalzed foms concenng the dffeent solve types. Unlke most numecal methods, the FIT dscetzes the followng ntegal fom of Maxwell s equatons, athe than the dffeental one: 39

49 A A B E. ds =. da, t A D H. ds = + J. da t A V V D. da = V B. da = ρ. dv (4.) (4.) In ode to solve these equatons numecally, a fnte calculaton doman s defned, enclosng the consdeed applcaton poblem. By ceatng a sutable mesh system, ths doman s splt up nto seveal small cubods, so-called gd cells. Ths fst o pmay mesh can be vsualzed n CST Mcowave Studo [9] n the Mesh Vew; howeve, ntenally a second o dual mesh s set up othogonally to the fst one. The Spatal dscetzaton of Maxwell s equatons s fnally pefomed on these two othogonally gd Fgue 4. Allocaton of the gd voltages and facet fluxes on the pmay and dual gds 4

50 systems, whee the new degees of feedom ae ntoduced as ntegal values as well. The electc gd voltages e and magnetc facet fluxes b ae allocated on the pmay gd G and the delectc facet fluxes d as well as the magnetc gd voltages h on the dual gd G ~ (ndcated by the tlde) as shown n Fg 4.. Now, Maxwell s equatons ae fomulated fo each of the cell facets sepaately as wll be demonstated n the followng. Consdeng Faaday s law, the closed ntegal on the left sde of Eq. (4.) can be ewtten as a sum of fou gd voltages wthout ntoducng any supplementay eos. Consequently, the tme devatve of the magnetc flux defned on the enclosed pmay cell facet epesents the ght-hand sde of the Eq. (4.), as llustated n Fgue 4. below. By epeatng ths pocedue fo all avalable cell facets, the calculaton ule can be summazed n an elegant matx fomulaton, ntoducng the topologcal matx C as the dscete equvalent of the analytcal cul opeato. Applyng ths scheme to Ampee s law on the dual gd nvolves the defnton of a coespondng dscete cul opeatoc ~. Fgue 4. Tme devatve of the magnetc flux defned on the enclosed pmay cell facet 4

51 Smlaly, the dscetzaton of the emanng dvegence equatons (Eq. (4.)) ntoduces dscete dvegence opeatos S and S ~, belongng to the pmay and dual gd, espectvely. As pevously ndcated, these dscete matx opeatos just consst of elements, and -, epesentng meely topologcal nfomaton. Fnally, the complete dscetzed set of the so-called Maxwell s Gd Equatons (MGE s) s obtaned. d Ce = b dt ~ d Ch = d + dt ~ Sd = q Sb = j (4.3) (4.4) Compaed to the contnuous fom of Maxwell s equatons, the smlaty between both descptons s obvous. Once agan, t should be mentoned that no addtonal eo has yet been ntoduced. Ths essental pont of the FIT dscetzaton pocess s eflected n the fact that mpotant popetes of the contnuous gadent, cul and dvegence opeatos ae stll mantaned n gd space: ~ ~ SC = SC = dv ot (4.5) ~ T ~ T C S = CS = ot gad (4.6) At ths pont, t should be mentoned that even the spatal dscetzaton of a numecal algothm could cause long tem nstablty. Howeve, based on the pesented fundamental elatons (Eqs. 4.5 and 4.6), t can be shown that the FIT fomulaton s not affected by such poblems, snce the set of MGE s (Eqs. 4.3 and 4.4) mantan enegy and chage consevaton [3]. 4

52 Fnally, the mssng mateal equatons ntoduce the nevtable numecal naccuacy due to the spatal dscetzaton. By defnng the necessay elatons between voltages and fluxes, the ntegal values have to be appoxmated ove the gd edges and cell aeas, espectvely. Consequently, the esultng coeffcents depend on the aveaged mateal paametes as well as on the spatal esoluton of the gd and ae summazed agan n coespondent matces as follows: D = ε E B = µ H J = σ E + J S d = M b = M j = M ε µ σ e h e + j S (4.7) (4.8) (4.9) Now, all matx equatons ae avalable to solve electomagnetc feld poblems on the dscete gd space. The fact that the topologcal and metc nfomaton ae dvded nto dffeent equatons has mpotant theoetcal, numecal and algothmc consequences [3]. Theefoe, the FIT fomulaton s a vey geneal method and can be appled to all fequency anges, fom DC to hgh fequences. 4.3 Implementaton of FIT Equatons nto Smulaton Softwae The smulaton method used fo the analyss s based on Fnte Integaton Technque capable of analyzng boadband stuctues, specfcally n the hgh-fequency ange. CST Mcowave Studo s a fully featued softwae package fo electomagnetc analyss and desgn n the hgh fequency ange [9]. It smplfes the pocess of nputtng the stuctue by povdng a poweful sold modelng font-end whch s based on the ACIS modelng kenel. Stong gaphc feedback smplfes the defnton of you devce even futhe. 43

53 Afte the component has been modeled, a fully automatc meshng pocedue (based on an expet system) s appled befoe the smulaton engne s stated. The smulatos featue the Pefect Bounday Appoxmaton (PBA method) and ts Thn Sheet Technque (TST) extenson, whch nceases the accuacy of the smulaton by an ode of magntude n compason to conventonal smulatos. Snce no method woks equally well n all applcaton domans, the softwae contans fou dffeent smulaton technques (tansent solve, fequency doman solve, egenmode solve, and modal analyss solve) whch best ft the patcula applcatons Tansent Solve The most flexble tool s the tansent solve, whch can obtan the ente boadband fequency behavo of the smulated devce fom only one calculaton un (n contast to the fequency steppng appoach of many othe smulatos). Ths solve s vey effcent fo most knds of hgh fequency applcatons, such as connectos, tansmsson lnes, fltes, antennas and many moe. A vey mpotant featue of the tansent solve s the excellent lnea scalng of the computatonal esouces wth stuctue sze. Cuently, moden pesonal computes allow the smulaton of stuctues wth a sze of up to oughly wavelengths. Ths smulato s equpped wth the new Multlevel Subgddng Scheme (MSS), whch helps to mpove the meshng effcency and thus, can sgnfcantly speed up smulatons, especally fo complex devces. Ths solve s used fo all the smulatons pefomed n ths thess. These ae some of the mpotant featues of the Tansent Solve: 44

54 Effcent calculaton fo loss-fee and lossy stuctues Boadband calculaton of S-paametes fom one sngle calculaton un by applyng DFT s to tme sgnals Calculatons of feld dstbutons as a functon of tme o at multple selected fequences fom one smulaton un Adaptve mesh efnement n 3D Plane wave exctaton (lnea, ccula o ellptcal polazaton) S-paamete symmety opton to decease solve tme fo many stuctues Calculaton of vaous electomagnetc quanttes such as: Electc felds, magnetc felds, suface cuents, powe flows, cuent denstes, powe loss denstes, electc enegy denstes, magnetc enegy denstes, voltages n tme and fequency doman Antenna fafeld calculaton (ncludng gan, beam decton, sde lobe suppesson, etc.) wth and wthout fafeld appoxmaton. Fafeld pobes to detemne boad band fafeld data at cetan angles Smultaneous pot exctaton wth dffeent exctaton sgnals fo each pot 4.3. Egenmode Solve Howeve, an effcent flte desgn often eques the dect calculaton of the opeatng modes n the flte athe than an S-paamete smulaton. Fo these cases, CST Mcowave Studo also featues an egenmode solve whch effcently calculates a fnte numbe of modes n closed electomagnetc devces. When nvestgatng hghly esonant stuctues such as naow bandwdth fltes, a tme doman appoach may become 45

55 neffcent, because of the slowly decayng tme sgnals. The usage of advanced sgnal pocessng technques (AR-fltes) povded by CST Mcowave Studo allows the speedng up of these smulatons by odes of magntude compaed to standad tme doman methods. These ae the mpotant featues of the Egenmode Solve. Calculaton of modal feld dstbutons n closed loss fee o lossy stuctues Adaptve mesh efnement n 3D Calculaton of losses and Q-factos fo each mode (dect o by usng a petubaton method) Automatc paamete studes usng the bult n paamete sweep tool Automatc stuctue optmzaton fo abtay goals usng the bult-n optmze Modal Analyss Solve Futhemoe, CST Mcowave Studo also contans a so-called modal analyss solve whch woks n combnaton wth the egenmode solve. Afte the modes of a flte have been calculated, ths vey effcent technque can be used to deve the S-paametes fo the flte wth lttle addtonal smulaton tme. These ae the mpotant featues of the Modal analyss solve. Boadband calculaton of S-paametes fom the modal feld dstbutons calculated usng the egenmode solve Re-nomalzaton of S-paametes fo specfed pot mpedances Calculaton of losses and Q-factos fo each mode (petubaton method) Automatc paamete studes by usng the bult-n paamete sweep tool 46

56 Automatc stuctue optmzaton fo abtay goals by usng the bult-n optmze Fequency Doman Solve The tansent solve becomes less effcent fo low fequency poblems whee the stuctue s much smalle than the shotest wavelength. In these cases t can be advantageous to solve the poblem by usng the fequency doman solve. Ths appoach s most effcent when only a few fequency ponts ae of nteest. The mpotant featues of the Fequency doman solve ae: Effcent calculaton fo loss-fee and lossy stuctues ncludng lossy wave gude pots Automatc fast boadband adaptve fequency sweep Use defned fequency sweeps Dect and teatve matx solves wth convegence acceleaton technques Pot mode calculaton by a D egenmode solve n the fequency doman Hgh pefomance adatng/absobng bounday condtons Peodc bounday condtons ncludng phase shft o scan angle Antenna fafeld calculaton (ncludng gan, beam decton, sde lobe suppesson, etc.) wth and wthout fafeld appoxmaton RCS calculaton Calculaton of SAR dstbutons Dscete elements (lumped esstos) as pots 47

57 Each of these solve s smulaton esults can then be vsualzed wth a vaety of dffeent optons. Agan, a stongly nteactve nteface wll help to quckly acheve the desed nsght nto a selected devce. The last, but not the least, outstandng featue s the full paametezaton of the stuctue modele, whch enables the use of vaables n the defnton of the selected devce. In combnaton wth the bult-n optmze and paamete sweep tools, CST Mcowave Studo s capable of both the analyss and desgn of electomagnetc devces and hence can solve vtually any hgh fequency feld poblem. 48

58 CHAPTER 5 RESEARCH APPROACH 5. Intoducton The pevous chapte ntoduced photodetectos, fundamentals of the EM waves, and pevous wok on MSM photodetectos. The basc smulaton equaton and the softwae used fo analyss wee also dscussed. Ths chapte pesents the methods and models used to smulate the MSM photodetecto to study mechansms fo nceased chage collecton effcency n sub-mcon scale gated photodetectos. The chage collected s a functon of the numbe of caes poduced n the actve egon of the detecto, whch n tun depends on the enegy avalable to ceate electon-hole pas n that egon. The semconducto photodetecto can be studed n tems of the enegy deposted by an electomagnetc wave of an appopate wavelength n the detecto actve egon. Besdes penetaton of the electomagnetc wave, t s also mpotant that the ncdent wave have enegy equal to o geate than the mateal bandgap. Ths chapte deals wth the electomagnetc feld analyss to explan the collecton effcency of gated photodetectos. Incease n the collecton effcency fo a gven wavelength of the ncomng electomagnetc sgnal s explaned. A nomal ncdent wave s consdeed fo the analyss of the S MSM detecto by lookng nto the tansmsson of the electomagnetc wave feld components as they tavel though the gated walls nto the devce actve egon. 49

59 5. Analyss of Squae and Wall lke Gatng Stuctues Fo slcon detectos the wavelength of nteest s nm, whch coesponds to the bandgap enegy. Thus, t s of nteest to know why the gatng stuctues ad n the deposton of moe enegy n the detecto suface, theeby nceasng cae geneaton and collecton. In the fst smulaton setup, a slcon substate was placed n between two alumnum contacts as shown n the Fg 5.. The dmenson of the S substate was µm 3. The devce was studed usng a plane wave pot (PWP) n the z decton wth a contnuous snusodal wave (λ = nm) as an exctaton sgnal as shown n Fg. 5.. One can also use a double exponental pulse wth an exctaton equaton gven n (5.) as the nput sgnal, but a contnuous sne µm µm µm Fg. 5. Smulated Stuctue wthout any gatngs wave has been chosen n ode to get a bette vew of the concept. Exctaton Functon = exp( E dtme) + exp( E dtme) (5.) Whee, and E ae constants. E E-felds wee calculated by placng the pobe at the nteface of the photodetecto. Fg. 5.3 shows the E-feld ampltude changes fo a detecto wthout any gatngs as the EM wave eaches the suface of the detecto. 5

60 Fg 5. Input Exctaton Sgnal (Contnuous Sne wave).5 Electc Feld (V/m) Tme (ps) Fg. 5.3 Incdent wave ampltude changes as t tavels though the suface of the detecto wthout any gatngs 5

61 In the second smulaton setup a slcon substate was placed n between two alumnum contacts ove whch the S sub-mcon scale wall-lke potusons wee placed, as shown n Fg The dmenson of the S substate was µm 3. The dmenson of the wall potuson was. µm 3. Anothe stuctue wth squae gatngs was also ceated fo compason as shown n Fg. 5.4 (b).. µm µm µm µm µm (a) µm. µm µm µm µm µm (b) Fg 5.4 Smulated Stuctue fo (a) walls and, (b) squae Gatngs The squae gatng s dmenson was. µm 3. The devce was studed usng a plane wave pot (PWP) n the z decton wth a contnuous snusodal wave (λ =. µm) as an exctaton sgnal. E-felds wee calculated by placng the pobes at the nteface between the substate and the S potusons (walls and squaes). 5

62 Fgue 5.5 shows the E-Feld ampltude changes fo a wall-lke gatng, and a detecto wthout gatng, as the ncdent electomagnetc wave tavels fom the estctve gatng egon to the substate egon. The objectve hee was to compute the E-felds nea the nteface of the two egons as the electomagnetc pulse penetates the potectve laye towads the nteo of the suface. Snce thee s a sudden change n the stuctue dmensons, a change n the mpedances of the two aeas would lkely affect the.5 Electc Feld (V/m) Walls None Tme (ps) Fg. 5.5 Incdent wave ampltude changes as t tavels fom the wall-lke gatng to the substate egons eflecton and tansmsson data. The ncease n ampltude s notceable fo the wall-lke stuctue as shown n Fg. 5.4(a). Fg. 5.6 shows the tansmtted wave ampltude usng the squae gatngs as compaed wth the wall-lke gatng and a detecto wthout any gatng. It can be seen that thee s hghe ampltude fo a squae-shaped gatng (Fg. 5.4 (b)) as compaed to the wall-lke stuctue. 53

63 A hghe tansmsson coeffcent value tanslates to moe enegy beng tansmtted to the substate egon fom the gatng egon. Thus, thee s hghe enegy tansmsson fo the wall-lke gatng stuctue as compaed to a detecto wthout the gatngs, as was epoted n expements [4]. Now, E τ = ncease n the tansmsson coeffcent means an ncease n the E t tansmtted wave ampltude. Ths can be explaned as follows: Consde two egons.5 Electc Feld (V/m) Squaes Walls None Tme (ps) Fg. 5.6 Incdent wave ampltude changes as t tavels fom the squae gatng to the substate egons (gatng wall as egon and actve aea as egon ) of the detectos as shown n Fg 5.7. Assumng nomal ncdence fom egon nto egon, the enegy caed by the wave s dstbuted between the eflected and tansmtted wave and can be chaactezed n 54

64 tems of the ncdent E feld ampltude, eflecton and tansmsson coeffcents, and the mpedances of the two meda. The ampltude of the tansmtted E-feld can be explaned n tems of consevaton of the tme aveaged powe (watts) due to E and H that ae cossng a gven suface A n the decton of the popagaton. z x Wall (aea = A ) Suface (S) wthout wall (aea = A ) Contact (Al) S S S t Fg 5.7 Sngle wall gatng on the actve egon used fo analyss Statng fom Eq. (.54), P = A S da = S A (5.) Whee, S s the aveage Poyntng vecto (watts/m ) deved fom the nstantaneous vecto S gven by S = E H (5.3) When the wave s tavelng fom a egon of coss sectonal aea A nto a egon of aea A one can wte elatonshps fo ncdent, eflected and tansmtted waves, 55

65 P (5.4) = S A S A and P = S (5.5) t A Hence at the nteface, one can wte, S S = S t P A P η P Γ = τ A η A η η τ Γ η τ η = τ = A A (5.6) If the mateals n egon ae the same as n egon (η = η ), and f we assume that A > A, t follows fom (5.6) that τ >. Thus, the ampltude of the tansmtted wave s geate than the ampltude of the ncdent wave, and an ncease n τ (wheeτ = E E t ) also tanslated to moe enegy deposton n the tansmtted o the actve egon of the detecto. Thus, the wall-lke stuctue poduces a lage numbe of electon-hole pas as compaed to an MSM detecto wthout the wall-lke gatng stuctue. It s expected that a squae gatng, whch poduces hghe values fo the tansmsson coeffcent, wll futhe mpove the collecton effcency of the detectos. Theefoe, the collecton effcency of the MSM detectos wth squae gatngs should mpove ove the wall lattce gatngs. 56

66 5.3 Analyss of Squae Gatngs wth Change n Aea Untl now, analyss shows mpoved effcency fo walls on the actve aea as compaed to a bae actve egon. It s also shown that a squae shaped gatng has bette effcency than a wall lattce. The next step s to detemne the numbe of squae lattce that should occupy the actve egon fo hghe effcency. Ths s done by changng the aea (A ), of the z x Squae (aea = A and A ' ) squae S extensons, whle keepng the detecto aea (A ) constant. Ths means that the pecentage of the aea coveed by A A ' A A the S extensons ove the detecto s changed. Howeve, the numbe Fg. 5.8 Font Vew of the Smulated Stuctue wth dffeent S extenson aeas of S extensons ove the detecto was kept constant (64). Smulatons wee caed out usng multtude numbe of aeas (A ). The font vew of the smulated stuctue s as shown n Fgue 5.8. The dmenson of the S substate was µm 3. The squae gatng dmensons wee vaed fom. µm 3 ( A ) to.5.5. µm 3 ' ( A ). As befoe, the devce was studed usng a plane wave pot (PWP) n the z decton wth a contnuous snusodal wave (λ =. µm) as an exctaton sgnal. The compason plot fo the esults s shown n Fgue

67 .5 Electc Feld Intensty (V/m) Aea A Aea A' Tme (ps).4.5 Fg 5.9 Incdent wave ampltude changes fo a wave tavelng though dffeent S extenson aeas Fom Fg. 5.9, t can be seen that as the aea of the S extenson s deceased, the E-Feld ntensty s nceased afte t eaches the steady state. Hee, A A < A ' (5.7) A ' Snce A s constant and A > A >, t follows fom Eq. (5.6), A τ > τ (5.8) ' A A t E Whee, τ =, E 58

68 Fg. 5. Plot of the E-Feld Vs Numbe of Spkes It follows fom Eq. 5.8 that a decease n the aea of the S extenson fom to nceases the tansmsson coeffcent, τ. It means, n ths case also, moe enegy s deposted n the tansmtted o the actve egon of the detecto. Hence, the electc feld ntensty s nceased because thee s a decease n the aea of each S extenson. Futhe smulatons wee done by changng the numbe of S extensons on the detecto. Howeve, the aea of the each S extenson s kept constant (.5.5. µm 3 ) because the hghest E-Feld has been obtaned wth ths confguaton. Smulatons wee caed out wth 36, 64,, 44, 96 and 56 numbe of squae S extensons. It can be seen fom Fg. 5. that as the numbe of S extenson nceases, the Electc Feld ntensty nceases. The sold lne shows the E-Feld wthout any S extensons. A ' A 59

69 Hence, the maxmum E-Feld s obtaned when the numbe of spkes s n between 9 and. Wth the ncease n numbe of spkes, the pecentage of aea coveed by the spkes on the detecto nceases. The ato of the aea coveed by the S spkes on the detecto to the aea uncoveed stats to ncease. Hence, the numbe of eflectons due to the adjacent S extensons nceases, whch account fo the ncease n the E-feld. But afte a cetan ctcal pecentage of the aea coveed by the S extensons on the detecto, the electc feld stats to decease as the numbe of eflectons decease. Hence, thee exsts a cetan ctcal ato of the aea coveed by the S spkes on the detecto to the aea uncoveed afte whch the E-feld stats to decease. 5.4 Analyss of Cone lke Gatng Stuctues The next step s to show an ncease n the E-feld ntensty f the stuctue s changed to a cone lattce fom a squae lattce.. µm Smulatons wee done µm wth a fxed numbe of S extensons (64), but ths tme, changng the µm R =.57 µm µm squae gatngs to cones. Fg. 5. Smulated stuctue wth cone gatngs The stuctue smulated s shown n Fg. 5.. The dmensons of the cone gatngs wee chosen such that the aea coveed by them on the detecto s same as the aea coveed by the squae gatngs on the detecto. 6

70 Fgue 5. shows the vaatons n the ampltude of E-Felds fo cone and squae type gatngs. It can be seen fom the fgue that the ampltude of the electc feld ntensty fo cone gatng s less than the electc feld ntensty fo a squae gatng..5 Electc Feld Intensty (V/m) Squae Cone Tme (ps) Fg. 5. Incdent wave ampltude changes as t tavels though cone and squae gatngs To explan the dffeence between the cone and the squae lattce, consde two cases fo each stuctue. Fst consde the popagaton of an EM wave at the suface of the gatng and agan consde the popagaton of EM waves at the nteface of the detecto and the gatngs. In case of suface nteactons, f a lght ay s ncdent on a suface, pat of the lght s eflected and pat may ente the second medum as the efacted ay, and may o may not undego absopton thee. Consde the case fo the cone and squae lattces shown n 6

71 Fg The amount of lght eflected depends on the ato of the efactve ndces of the two meda. θ θ θ A A θ A A (a) (b) Fg 5.3 Analyss of (a) Cone and, (b) Squae gatng stuctues Recall fom Chapte, that f the wave tavels fom one medum to anothe (n ths case fom a to S), we can apply an mpotant law, called Snell s law, whch states that the poduct of the efactve ndex and the sne of the angle of ncdence of a ay n one medum s equal to the poduct of the efactve ndex and the sne of the angle of efacton n a successve medum. Algebacally, ths can be wtten as η snθ = η snθ, (5.9) whee, η, η ae the two values of efactve ndex and θ, θ ae the angles of ncdence and efacton. The ncdent ay, the efacted ay, and the nomal to the bounday at the pont of ncdence all le n the same plane. 6

72 Geneally, the efactve ndex of a dense tanspaent substance s hghe than that of a less dense mateal; that s, the speed of lght s lowe n the dense substance. So, f a ay s ncdent on the suface, then a ay enteng a medum wth a hghe efactve ndex, t wll be bent towads the nomal, and a ay enteng a medum of lowe efactve ndex wll be bent away fom the nomal. Rays ncdent along the nomal ae eflected and efacted along the nomal. Fg 5.3 shows the font vew of the popagaton of the EM wave at the suface of the cone and squae gatngs. The aea of the detecto, A and the aea coveed by the gatng on the detecto, ae same fo both the cone and the squae. One can see fom Fg. 5.3 A (a) that as the electomagnetc wave s ncdent on the suface of the cone at an angle θ, the efacted ay tends to move towads the nomal. In ou case η =, η =3.59. Hence, the ato η / η>. If t s assumed that the angle of ncdence, θ s constant, t follows fom Eq. (5.9) that θ > θ. Theefoe, the angle of efacton θ tends towad the nomal and also away fom the nteface, afte whch, the wave may leak outsde the cone gatng. Hence, not all the enegy of the EM wave cosses the nteface. Ths accounts fo the lesse effcency of a cone shaped gatng. On the othe hand, fo the case of a squae gatng, t s a completely dffeent stuaton. As the EM wave s ncdent on the suface of the squae gatng, the angle of efacton θ bends towad the nomal afte whch t tavels towad the nteface as shown n Fg. 5.3 (b). Ths accounts fo the ncease n the numbe of chage caes at the nteface. Hence, the oveall chage collecton effcency of the detecto wth squae gatngs 63

73 nceases. Thus, one can conclude that even though the nteface aea of the cone and the squae lattce may be the same, but the amount of enegy cossng the nteface s much less fo the cone. 5.5 Analyss of Squae Gatngs wth Claddng Howeve, thee s a possblty that the EM wave leaks away fo the case of squae gatngs, but s less than that fo a cone gatng. Futhe smulatons wee done wth dopng aound the gatngs to pevent the leakage of the chages and fo the hghe powe tansmsson nto the actve egon theeby enhancng the collecton effcency due to the eflectons fom the suface.. µm µm µm µm Fg 5.4 Smulated Stuctue wth dopng aound the squae gatngs Fg. 5.4 shows the stuctue smulated wth all the dmensons of the detecto suface and the gatngs beng the same as that of the eale case, except that the squae gatngs ae doped wth a mateal of lowe efactve ndex than S. A thn laye of dopng mateal s doped aound the squae gatngs to ncease the chage collecton effcency. 64

74 .5 Electc Feld Intensty (V/m) Wthout Coatng Wth Coatng Tme (ps) Fg. 5.5 Incdent wave ampltude changes as t tavels though squae gatngs wth and wthout dopng (coatng) Fg. 5.5 shows the smulated esults fo the case of squae gatngs wth and wthout dopng. It can be seen fom the fgue that thee s an ncease n the ampltude of the electc feld ntensty fo the case of the squae gatngs wth dopng than the case wthout dopng. Befoe gong nto the detals, t s of mpotance to know the bascs of fbe optcs. Fst, a few equatons elated to optcal fbes ae deved and then appled to ths study. Fg 5.6 shows how the waves ae guded along optcal fbes. Lght can be guded though thn delectc ods made of glass o tanspaent plastc, known as optcal fbes. Because the lght s confned to tavelng wthn the od, the only loss n powe s due to eflectons at the sendng and ecevng ends of the fbe and absopton of the fbe 65

75 mateal (as t s not a pefect delectc). The coe hee s S wth a efactve ndex η S. It s suounded by a claddng wth efactve ndex, η c. whee, η c < ηs. Also, η s the efact ve ndex of the medum suoundng the fbe (n ou case t s =). When the wave s ncdent at an angle θ, pat of the wave s eflected and pat of th e wave s tansmtted nto the coe wth an angle, θ. Ths wave s then ncdent to the claddn g at an angle, θ and agan pat of t s tansmtted wth an 3 angle θ nto the claddng. The case of t θ θ θ 3 θ t c c l l η a a η d d d d η c n n g Coe g η c nteest hee s when θ t = π /, whee all the waves ae eflected back nto the coe and no enegy s tansmtted nto the claddng. To satsfy ths condton of total ntenal eflecton, the ncdent angle θ 3 n the coe must be equal to o geate tha n the ctcal Fg. 5.6 Gudng of waves n a squae gatng angle, θ c fo the wave n the coe medum ncdent upon the claddng medum. Recall Eq. (5.9) and substtutng θ t = π / we have, ηc sn θ c = η S (5.) 66