2. Dispersion in the Planar Waveguide

Transcription

1 Chapt.2_2 Words Dispersion diagram( 色散图 ), modal/intermodal dispersion( 模间色散 ), intermodal coupling( 模间耦合 ), intramodal dispersion( 模内色散 ), penetration depth( 渗透深度 ), mode field distance(mfd, 模场距离 ),

2 2. Dispersion in the Planar Waveguide Dispersion Diagram From the waveguide condition, 2 π n1 (2 a) cos θm φm = m, here λ tan ( 1 φ ) 2 m = sin 2 θ m cos π m = 0,1, 2, (2 2) n n 2 θ m (1 23)

3 Waveguide Dispersion Diagram Dispersion diagram: n 1, n 2, a and λ β m for each ω and each m the ω vs. β m characteristics as in Fig ω cut-off ω Slope = c/n 2 TE 2 TE 1 Slope = c/n 1 2π c here ω =, λ 2π n1 βm = sinθm λ TE 0 β m Figure 2.10 The slope dω /dβ m (dω /dk) at any ω is the group velocity v g. The cut-off frequency ω cut-off the cut-off condition λ c. v g at one frequency changes from one mode to another; For a given mode it changes with the frequency ---- v g (ω). Schematic dispersion diagram, ω vs. β for the slab waveguide for various TE m. modes. ω cut 杘 ff corresponds to V = π/2. The group velocity v g at any ω is the slope of the ω vs. β curve at that frequency.?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

4 If Intermodal Dispersion Modal dispersion (or intermodal dispersion): In multimode operation, well above ω cut-off, the lowest mode (m = 0) has the slowest group velocity ~ c/n 1 ; the highest mode has the highest group velocity ~ c/n 2. the modes take different times to travel the length of the fiber broadening signal as in Fig.2.7. Δτ is the propagation time difference between the fastest & slowest modes over a distance L, the modal dispersion is defined L L Δτ = v v g min g max gmin gmax (2 11) where v is the minimum group velocity of the slowest mode, v is the maximum group velocity of the fastestmode.

5 Intermodal Dispersion Since v c / n, v c / n Δτ Thus L gmin 1 gmax 2 n n c 1 2 In general, intermodal dispersion by this estimate (2 12) (2 12) is not as high as indicated due toan" intermode coupling". In optoelectronics, the spread Δτ 1/2 between the half intensity points. But when many modes are present, take Δτ Δτ. 1/2

6 Intramodal Dispersion Waveguide dispersion: The higher λ (lower ω) is, the greater the penetration of the field into the cladding as in Fig.2.11 a greater portion of the light energy is carried by the cladding ---- phase velocity is higher. Longer wavelengths propagate faster even by the same mode (v g (ω) in Fig2.10) --- the guiding properties of the dielectric structure --- the frequency dependence of the index.

7 Intramodal Dispersion y y Cladding λ 1 > λ c λ 2 > λ 1 v g1 Core v g2 > v g1 ω 1 < ω cut-off ω 2 < ω 1 E(y) Cladding Figure 2.11 The electric field of TE 0 mode extends more into the cladding as the wavelength increases. As more of the field is carried by the cladding, the group velocity increases.?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

8 Intramodal Dispersion Material dispersion: n 1 will also depend on the λ modify the ω -β m behavior in Fig The change in the group velocity of a given mode due to the n - λ dependence also gives rise to the broadening of a propagating light pulse. Intramode dispersion: Both waveguide and material dispersion act together to broaden a light pulse propagating within a given mode (combined dispersion).

9 2.8 Dielectric slab waveguide Consider a slab dielectric guide that has a core thickness(2a)of 10 μm, n 1 =3,n 2 = 1.5. Solution of the waveguide condition in Example 2.1 gives the mode angles θ 0 and θ 1 for the TE 0 and TE 1 modes for selected wavelengths as summarized in the table below. For each wavelength calculate ω and β m and then plot ω vs. β m. On the same plot show the lines with slopes c/n 1 and c/n 2. Compare your plot with the dispersion diagram in Fig λ, Questions and Problems μm θ θ

10 Questions and Problems 2.9 Dielectric slab waveguide Consider a planar dielectric guide with a core thickness 10 μm, n 1 =1.4446, n 2 = Calculate the V-number, the mode angle θ m for m = 0 (use a graphical solution, if necessary), penetration depth, and mode field distance (MFD = 2a +2δ), for light wavelengths of 1.0 μm and 5 μm. What is your conclusion? Compare your MFD calculation with 2ω o =2a(V+1)/V.

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