Planar dielectric waveguides

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Wesley Townsend
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1 Plaar dieletri waveguides Abstrat: A optial waveguide is a physial struture that guides eletroageti waves i the optial spetru. They are used as opoets i itegrated optial iruits, as the trasissio ediu i log distaes for light wave ouiatios, or for bioedial iagig. We a lassify the waveguide aordig to differet ethods. Aordig to the strutures: plaar, strip, or fiber waveguides, ode struture: sigle-ode, ulti-ode, refrative idex distributio: step or gradiet idex ad aterial: glass, polyer, seiodutor. Here we disuss the plaar dieletri waveguide speifially fro the ofiguratio, waveguide ode, field distributio, dispersio relatio ad group veloity aspets. : Cofiguratio Fig. shows the ofiguratio of a typial plaar dieletri waveguide. A slab of dieletri aterial, alled fil or ore, surrouded by edia of lower refrative idexes, alled over ad subtrat as the upper ad lower, respetively. Fig. (Plaar dieletri waveguide ofiguratio. The width of the slab is d ad refratio idex is, ad the over ad subtrat have sae refratio idex.) A light ray a be guided iside the slab by total iteral refletio i the zigzag fashio. Oly ertai refletio agle θ will ostrutively iterfere i the waveguide ad hee oly ertai waves a exist i the waveguide (this will be disussed ore i setio waveguide odes). Case : θ saller tha opleet of the ritial agle θ < θ π si ( / ) os ( / ) ase / Total iteral refletio will happe at the boudaries. The the rays a travel i z diretio by

2 bouig betwee the slabs surfaes without loss of eergy (figure showed i the right of Fig.). Ad we also assue that all the aterials are lossless. Case : θ larger tha opleet of the ritial agle θ > θ π si ( / ) os ( / ) ase / Total iteral refletio a ot happe at the boudaries. The rays will lose a portio of their power at eah refletio, ad evetually they will vaish. I this paper, we oly osider syetri plaar dieletri waveguide, whih is the over ad subtrat have the sae refratio idex. : Waveguide odes Beause oly ertai refletio agle θ are allowed. We eed self osistey oditio to fid θ whih a survive i the waveguide, ad eas th ode. Assuptio The field i the slab is i the for of a oohroati TEM plae (Eletri wave is osillatig perpediular to iidet ad refletio plae, here is x diretio), ad wave boues i ase situatio disussed i setio of ofiguratio. /, / The wave vetor is K, havig K osθ., K K siθ, K K x y z Self-osistey oditio: the phase shift betwee the two waves ust be or a ultiple of π A wave should reprodue itself after eah roud trip, otherwise they will have phase shift ot equal to a ultiple of π. I oe roud trip, the twie refleted wave lags behid the origial wave by a distae AC AB d *siθ, as i FIG. At the dieletri boudary, eah iteral refletio will itrodue a phaseφ r. π d siθ φr π Eq φ r, depeds o the agle θ ad the polarizatio of the iidet wave. Aordig to the TE wave refletio phase shift ad θ π / θ, θ π / θ, θ is the opleet agle of θ. φr si θ ta si θ Eq As θ varies fro toθ, φ r varies fro π to. Substitute Eq ito Eq ad we a get self

The RHS ad LHS are plotted versus siθ. Τhe itersetio poits, arked by filled irles, deterie siθ.

3 osistey oditio for TE odes. d π si θ ta( π siθ ) si θ Eq 3 Fig : (Graphial solutio of Eq3 to deterie the boue agle θ of the odes of a plaar dieletri waveguide. The RHS ad LHS are plotted versus siθ. Τhe itersetio poits, arked by filled irles, deterie siθ. Eah brah of the ta or ot futio i the LHS orrespods to a ode. I this plot si 8( / d) Propagatio ostat θ ad the uber of ode is M9. ) The wave vetor with agle θ have the opoets (, K y K siθ, K osθ ). The z opoet is the propagatio ostats. It shows i Fig. 3 β K osθ Eq 4 Fig. 3 (The boue agles θ ad the orrespodig opoets of the wave vetor of the

4 waveguide odes are idiated by dots. The propagatio ostat lies betwee K ad K.) Nuber of odes Aordig to Eq3 ad beausesiθ siθ, we a get the uber of waveguide s odes. siθ M ; / d d M NA; NA NA is the uerial aperture. Ad the uber of ode is ireased to the earest iteger. I a dieletri waveguide there is at least oe TE ode, sie the fudaetal ode is always allowed. Sigle ode waveguide: whe / d > siθ, oly oe ode is allowed. This ours whe the slab is thi eough or the wavelegth is suffiietly log. At this situatio, there is o utoff wavelegth. But eah other ode, higher tha, has utoff wavelegth. The oditio for sigle ode operatio is that ν >, ν ω / π Eq 6 NA d This a be show i Fig 4. ν Eq 5 Fig 4 (Nuber of TE odes as a futio of frequey.) 3 Field distributio Iteral field There are two oposed TEM plae waves travelig at agles θ ad -θ with z axis with wave vetor opoets, ± K siθ, K osθ ). They have the sae aplitude ad (

5 phase shift π at the eter of the slab the we a get eletri field oplex aplitude E ( y, a u ( y)exp( jβ Eq 6 x β is propagatio ostat., is a ostat, ad a siθ os(π y),,,4... u( y), d / y d / siθ si(π y),,3,5... Eq 7 Note: the field does ot vaish at the boudary. If the iterfae of the boudaries are irrors, the the exteral field are zero. Exteral field The exteral field ust ath the iteral field at all the boudary poits y ± d/. So it ust vary with z as exp( β. Substitute E ( y, a u ( y)exp( jβ ito Helholtz j x equatio ( + K ) E ( y, x, we a get exp( γ ), y > d / u( y) ; exp( γ ), y < d / os θ γ K os θ Eq 8 γ is the extitio oeffiiet. Ad this wave is alled evaeset wave. As the ode uber ireases, θ ireases, ad γ dereases. Higher order odes therefore peetrate deeper ito the over ad substrate. It shows i Fig. 4 Fig 4 (Field distributio for TE guided odes i a dieletri waveguide.) The field distributio of the lowest order TE ode is siilar i shape to that of the Gaussia bea, but guided light does ot spread i the trasverse diretio as it propagates i the axial

diretio. I a waveguide, the tedey of light to diffrat is opesated by the guidig atio of the ediu. It shows i Fig. 5 Fig. 5 (a) Gaussia bea i a hoogeeous ediu; b) Guided ode i a dieletri waveguide.

6 diretio. I a waveguide, the tedey of light to diffrat is opesated by the guidig atio of the ediu. It shows i Fig. 5 Fig. 5 (a) Gaussia bea i a hoogeeous ediu; b) Guided ode i a dieletri waveguide. ) 4 Dispersio relatio ad group veloities Chage Eq i ters of β ad usig the futio that K y ( ω / ) β ω d β φ r + π Eq 9 Also Eq s for a be haged to the dispersio relatio. ta d ( ω β π ) Rewrite Eq ito paraetri for, β ω / ω / β Eq ω ω ω / π ( + ta π / dna ), β ω /, Eq is the effetive refrative idex defied i Eq, ad ω is the ode utoff agular frequey. We a fid out the effet of a stroger ofieet of waves of shorter wavelegth i the ediu of higher refrative idex. We should ote that higher order odes travel loger distae i the waveguide tha do lower order odes. Thus for light lauhed at the sae tie, the tie of arrival at the far ed of the wavelegth will deped o the path take. This results i a spread i tie of arrival. This is pulse broadeig. Modal dispersio: I propagatio through a ultiode waveguide, optial pulses spread i tie sie the odes have differet veloities. Group veloity dispersio (GVD): The group veloity is obtaied fro the dispersio relatio by deteriig the slope v dω / dβ for eah of the guided odes. Ad the group veloity of the allowed odes rage fro to a value slightly below. I a sigle ode waveguide, a optial pulse spreads as a result of the depedee of the group

7 veloities o frequey. It happes i hoogeeous aterials by virtue of the frequey depedee of the refrative idex of the aterial. Moreover, it ours i waveguides eve i the absee of aterial dispersio. With a soure with a rage of wavelegths, there will be a rage of group veloities. It results fro the guidig properties of the waveguide ad has othig to do with the frequey depedee of the refrative idex. Loger wavelegth has ore eergy i the laddig ad thus travels faster. Eah ode has a partiular agular frequey at whih the group veloity hages slowly with frequey the poit at whih v reahes its iiu value so that its derivative with respet to ω is. At this frequey, the GVD oeffiiet is ad pulse spreadig is egligible.

Solution 2. n n. n n R. n n. R n + = 1. n n = + n n. n n. n n. Insert R N = 0.95 and solve, + R n. 1 ln. =14.58 i.e. 15 pairs are needed.

Solution 2. n n. n n R. n n. R n + = 1. n n = + n n. n n. n n. Insert R N = 0.95 and solve, + R n. 1 ln. =14.58 i.e. 15 pairs are needed. .9 Dieletri irror Cosider a dieletri irror that is ade up of quarter wave layers of GaAs with H.8 ad AlAs with L.9, both aroud 5. The GaAs-AlAs dieletri irror is iside a vertial avity surfae eittig laser