Lect. 4 Waveguides (1)

Transcription

1 Lect. 4 Waveguies (1) - Waveguie: Confines an guies EM waves Metallic, Dielectric, Plasmonic - We are intereste in ielectric waveguie Total internal reflection b refractive inex ifferences n A B C D n k 0 β h Silicon Photonics (015/)

2 Lect. 4 Waveguies (1) Total internal reflection b refractive inex ifferences n A B C D n k 0 β h - Conitions for guiance - Characteristics of guie light in a ielectric waveguie: Moe, Effective inex, Confinement factor - How to make ielectric waveguies Silicon Photonics (015/)

3 - Conitions for guie EM waves Lect. 4 Waveguies (1) n =/ =0 n =-/ - Governing equations for EM waves B E t D H J t D D E B 0 B H Silicon Photonics (015/) E t H t E H EM Wave Equations Governs light propagation

4 Lect. 4 Waveguies (1) - Solutions for Wave Equations E E t Plane-wave solution: E E j ( tk x( k ) E ) 0e xe 0 e j( tk) E t j( t k) x ( ) E0e k k T k T 1 Spee of light! Silicon Photonics (015/)

5 Lect. 4 Waveguies (1) How oes the plane-wave solution look like? ( ) For phsical representation, Re j tk xe0e xe0cos( t k) At t=0 At t>0 Silicon Photonics (015/) W.-Y. Choi

6 How about H-fiel? E E t Lect. 4 Waveguies (1) E xe From Maxwell s Equations, Direction of propagation? Direction of E, H fiels? Spee of propagation? 0 e j ( t k ) H E0e j( tk) E H 0 0 (377for vacuum) Silicon Photonics (015/)

7 Lect. 4 Waveguies (1) How oes the plane-wave solution look like? Silicon Photonics (015/)

8 Lect. 4 Waveguies (1) When a wave is propagating into + irection: - irection: + irection: An irection? Plane wave solutions j( t k) e j( t k) e j( t k) e jk jt jkxx e e e e jk e j( t k R) k xkx k k R xx k, k: irection of propagation Silicon Photonics (015/)

9 Lect. 4 Waveguies (1) - Mathematical solutions for guie EM waves n n E t =/ =0 =-/ E (,, ) ( t,, ) t j t Assuming E,, t E, e, E k ( ) E 0, where k ( ) ( ) k( ) nk0 for ; claing k( ) nk 1 0for ; core Silicon Photonics (015/)

10 Lect. 4 Waveguies (1) n =/ =0 n =-/ Consier TE Solution (or E having onl x-component) j E(, ) xe ( ) e E( ) Then, ( k ( ) ) E( ) 0 => Eigen value equation. Solve for an E( ) k( ) 0 in core => E ( ) ~ sin( k) or cos( k) with k ( nk) 1 0 k ( ) 0 in claing => E( ) ~ exp( ) or exp(- ) with ( n k ) 0 Silicon Photonics (015/)

11 Lect. 4 Waveguies (1) n =/ =0 n =-/ Solutions : E( ) Aexp( ) Bexp( ) : E( ) Csin( k) Dcos( k) : E( ) Eexp( ) Fexp( ) Here, A=0 an F=0. For eas analsis, ivie the solutions into even an o solutions Silicon Photonics (015/)

12 Lect. 4 Waveguies (1) n =/ =0 n =-/ Even Solutions : E( ) Bexp( ) : E( ) Dcos( k ) : E( ) Bexp( ) ( E B) Appl bounar conitions: E( ) E( ) an are continuous at Bexp( ) Dcos( k ) (1) Bexp( ) kdsin( k ) () () k tan( k ) (1) Silicon Photonics (015/)

13 Lect. 4 Waveguies (1) n =/ =0 n =-/ O Solutions : E( ) Bexp( ) : E( ) Dsin( k ) : E( ) Bexp( ) ( E B) Appl bounar conitions. E( ) E( ) an are continuous at Bexp( ) Dsin( k ) (1) Bexp( ) kdcos( k ) () () kcot( k ) k tan( k ) (1) Silicon Photonics (015/)

14 n n Silicon Photonics (015/) Lect. 4 Waveguies (1) =/ =0 =-/ What o these mean? For graphical analsis, o following normaliation. Let X k, Y Then, Y X tan X for even Y X tan( X ) for o But X Y ( k ) [( nk ) ( n k ) ] k0 ( n1 n ) r Even: k tan( k ) O: k tan( k ) Determine k an that satisf above conitions. Plot these on X-Y plane.

15 Lect. 4 Waveguies (1) Y Y Xtan X : even moe m=1 m= m=3 Y X tan( X / ) : o moe X Y k0 ( n1 n ) r Observations: - Points where circle an tangent curves intersect are solutions moe 3 X k With larger r (larger, smaller, larger -n ), more moes exist in the waveguie Silicon Photonics (015/)

16 Lect. 4 Waveguies (1) E() profile: =1.5, n =1.495, =10m, =1m TE 1 TE Silicon Photonics (015/)

17 Lect. 4 Waveguies (1) E() profile: n1=1.5, n=1.495, =10 m, =1 m TE1 Silicon Photonics (015/) TE3 TE

18 Lect. 4 Waveguies (1) - Effective Inex: N = /k 0 Different moes have ifferent effective inices: Moal ispersion - Confinement factor: Power insie core Total Power E( ) E( ) For higher moes, how oes change? Silicon Photonics (015/)

19 Lect. 4 Waveguies (1) - Other polariation? n =/ =0 n =-/ TE solution (or E having onl x-component) was assume j E(, ) xe ( ) e TM solution exists H(, ) x H( ) e j In general, TM solution has ifferent guie-wave characteristics Silicon Photonics (015/)

20 Lect. 4 Waveguies (1) Issues for practical waveguies - Precise control of imension an refractive inex - Low loss at esire - Mass prouction possible - Integration esirable - Electrical control of refractive inex an/or absorption Materials use for waveguies - Silica (SiO with Ge oping) Optical fiber - Dielectric materials: LiNbO 3 with Ti oping - Semiconuctors: GaAlAs, InGaAsP, Si/SiO Silicon Photonics (015/)

21 Lect. 4 Waveguies (1) Optical Fiber: Circular ielectric waveguie mae of silica (SiO ) Claing Core SiO :Ge r Fiber axis n n The step inex optical fiber. The central region, the core, has greater refractive inex What than is special the outer about region, fiber? the claing. The fiber has clinrical smmetr. We use the coorinates r,, to represent an point in the fiber. Claing is normall - Extremel much thicker low loss: than 0.B/km shown. - Can be ver long: 100 s of km?1999 S.O. Kasap, Optoelectronics (Prentice Hall) Silicon Photonics (015/)

22 Lect. 4 Waveguies (1) Loss in fiber Loss (B/km) Raleigh scattering OH - absorption peaks 1310 nm 1550 nm Lattice absorption Wavelength () Silicon Photonics (015/)

23 Lect. 4 Waveguies (1) LiNbO 3 waveguie Coplanar strip electroes V(t) Thin buffer laer Polarie input light L E a LiNbO 3 LiNbO 3 EO Substrate x Waveguie Cross-section Integrate tranverse Pockels cell phase moulator in which a waveguie is iffuse into an electro-optic (EO) substrate. Coplanar strip electroes appl a transverse fiel E a through the waveguie. The substrate is an x-cut LiNbO 3 an tpicall there - Use is a thin for ielectric high-spee buffer optical laer (e.g. moulator ~00 nm thick SiO ) between the surface electroes an the substrate to separate the electroes awa from the waveguie.?1999 S.O. Kasap, Optoelectronics (Prentice Hall) Silicon Photonics (015/)

24 Lect. 4 Waveguies (1) Example of Si/SiO waveguie on SOI wafer fabricate with Si technolog (Rib/Rige Waveguie) (Strip/Channel Waveguie) What affects the characteristics of Si/SiO waveguies? Sie, scattering loss Silicon Photonics (015/)