Analysis Methods for Slab Waveguides

Size: px

Start display at page:

Download "Analysis Methods for Slab Waveguides"

Charlotte Washington
5 years ago
Views:

1 Aalsis Methods for Slab Waveguides Maxwell s Equatios ad Wave Equatios Aaltical Methods for Waveguide Aalsis: Marcatilis Method Simple Effective Idex Method Numerical Methods for Waveguide Aalsis: Fiite-Elemet Methods (FEM) Fiite-Differece Methods (FDM) Beam Propagatio Methods (BPM)---ouiform i the propagatio directio

2 Optical Slab Waveguides film (core laer) > superstrate (upper claddig laer) > substrate (dow claddig laer) ----EM waves i the waveguide core propagate alog the z.

3 Zig-zag wave Model Propagatio costat β : β = k si θ ( 7) sice ksiθ β k c (critical agle θc = si ) therefore β / k < (guided modes) Self-cosistec coditio : thelight phase after oe roud trip (betwee poits A ad B) t kcosθ Φ Φ = mπ m= (8) where Φ ad Φ are phase shifts acquired at the boudaries uder total iteral reflectio.

4 ( ) θ = β sice kcos k / k therefore from( 8) 4U Φ Φ = mπ m= ( 9) ( the Trasverse Resoace Coditio= = the Eigvalue Equatio ) Goos-Hache shift: Evaescet fields i the substrate ad superstrate ca be take ito accout i zig-zag wave model b cosiderig Goos-Hache shifts at the waveguide boudaries Effective waveguide thickess : t = t+ x + x ( ) eff w where x = x = for TE modes. w

5 Maxwell s Source Free Equatios E =jωμh H = jωε E ( ) ( ) where E ( x z) H ( x z) are electric ad magetic fields; ω is optical agular frequec; με is permeabilit ad permittivit of medium ( μ= μ ). ε ote : = ( x ) ε ωμ = k here wave umber k = π /. ; ε λ

6 For guided waves alog z axis : jβ z E( xz ) = E( xe ) jβ z H( xz ) = H ( x ) e where β is propagatio costat a ( ) ( 4) d z-directed compoet of k. For guided waves : = z jβ ( 5) For slab waves: ( ) ~ ( 6) ( 6) i.e. E E E H H H are fuctios of x ol so from = x z x z we obtai two equaios for TE & TM modes.

7 d TE modes (trasverse electric field modes E =E ) dx E with E + = H H x [ ( x) k β ] E x z ( 7) = E = H = z β = ωμ = jωμ E de dx. TM modes (trasverse magetic field modes H =H ) d dh β k H + dx dx = ( x) with E E E x z ( 8) = H = H = = = x β ωε jωε z H dh dx

8 / β U = kt k / β W = kt Defie : k / β W = kt ( 9) ( ) ( ). k ( β / k = eff is mode idex or effective idex) the solutio of Eq.( 7) & ( 8) E x Aexp W x > t t x = A cos U + ψ x t x Aexp W x < t t ( ) t H W A cos( U + ψ ) exp ( x t) x t t > x = A cos U + ψ x t t W A cos( U ψ ) exp ( x + t) x<t t ()

9 de from ( ) with E cotiuous at x= t/ t dx obtai Eigevalue Equatio: 4U Φ m TE TE Φ = π m = ( 4) / β TE k where W Φ = ta = ta U β mk β Φ TE / β W k ta U β = ta = TEm k k waves ( m is mode order)

10 ( 5) Eigevalue Equatio from (-) with cotiuous at / obt i : a 4 TM TM m d x t t U m x H H d π = = Φ = Φ / / ta ta ta ta where TM TM W k U k W k U k β β β β γ γ γ γ = = Φ Φ = = = =

11 Dispersio Relatio Normalizatio frequec: ν = kt( ) / ( 6) Normalizatio propagatio costat: b = β / k ( ) mode cutoff : β / k = or b= ( 7) ( 8)

12 The eigevale equatio becomes from ( 4) ( 6) ( 7) ( ) ( / ) ( b) ( b a) ( b) ν b = mπ + ta γ b/ + ta γ + / where γ γ a = = = / for TE mode ( 9) for TM mode for TE mode for TM mode for both TE ad TM modes Note: The eigevalue equatio ca be solved b umeric computer; b vdiagrams Dispersio Relatio Figure (ext page)

13 Useful Iformatio Uiversal dispersio curves for the TE mode for a three-laer slab waveguide whose eigevalue equatio is give i Eq. (-9) : a = b = ( β / k ) ν = kt( ). /

14 Cutoff Coditios Settig b = (mode cutoff) from ( 9) : Normalizatio frequec Table.4 ν c = mπ + γ a ta ( ) Modes Rage of β/k Mode Fields Guided β/k (discrete) Substrate β/k < (cotiuous) Superstrate < β/k < (cotiuous) Mode Tpes Oscillator i film; Evaescet i both substrate ad superstrate Oscillator i both film ad substrate; Evaescet i superstrate Oscillator i film substrate ad superstrate

15 Power Flow i Slab Waveguides Theguided modes carr power i the z directio ol. the z-compoet of the Potig vec tor: * * S = ExH H xe () * β S = EH x = E ( ) ωμ P TE for TE modes β + = E dx ( ) ωμ (power desit per uit legth i the ) for TM modes * β S = ExH = E ( 4) ωε P TM (power desit per uit legth i the ) β + H = dx ( 5) ωε

16 Example. : Please fid the total umber of TE ad TM modes guided b a mm thick slab waveguide i air show i Fig.E.where the slab idex is.5 ad the substate idex is.. The vacuum wavelegth is 6 m. Solvig: From the eigevalue equatio m= 4 Φ Φ π [ U ] ( ) At cutoff β / k = & Φ = b= U k / / / TE TM = t Φ = ta ta Φ = 8π t the TE modes: m= ( ) π λ / ta Fig.E. /

17 with t = mm =.5 =. =. we fid : m TE TM m = { 4 = } 6.8 t 64.6 TE m =.5. ta 9 TM m =.99 (i.e.tmtm) 64.xx / If reduced to μm (thikof opticalfibre). (i.e.te TE TE ) therefore the mm thick waveguide supports 65 (/μm) TE modes ad 65 (/ μm) guided TM modes. guided

18 Exercise Oe.Startig from the wave equatios (-7) (-8) derive the eigevalue equatios for the TE ad TM modes of a smmetric three-laers lab waveguide. The idex i the slab is surroudig medium is. The thickess of the ad the idex i the ( ) the free-space wave umber. [ Note:The derivatio ca be simplified whe the smmetr boudar coditio is applied at the cetre of the structure; slab is t; Verif our results b reducig the eigevalue equatio for a geeral asmmetric slab waveguide preseted i the lecture; Determie also the cutoff ν values where ν = kt with k = π / λ de E x t a b a b m dx are all cotiuous at = ; ta( ) = = ta ( ) + π ]

19 See ou ext time!

Ray Optics Theory and Mode Theory. Dr. Mohammad Faisal Dept. of EEE, BUET

Ray Optics Theory and Mode Theory. Dr. Mohammad Faisal Dept. of EEE, BUET Ray Optics Theory ad Mode Theory Dr. Mohammad Faisal Dept. of, BUT Optical Fiber WG For light to be trasmitted through fiber core, i.e., for total iteral reflectio i medium, > Ray Theory Trasmissio Ray