We first discuss the case of the planar waveguide in terms of the carrier compensation process. If the waveguide core layer is assumed homogeneous, then the amount of free-carrier compensation necessary for waveguiding, that is, at least for the fundamental transverse electric mode (TE_{0}) to propagate, can be calculated. This cutoff condition for TE_{0} to propagate is [1]

$\frac{\mathrm{2}\mathit{\pi}\mathit{d}}{{\mathrm{\lambda}}_{\mathrm{0}}}\sqrt{{\mathit{n}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{-}{\mathit{n}}_{\mathrm{2}}^{\mathrm{2}}}\mathrm{\ge}\mathrm{arctan}\frac{{\mathit{n}}_{\mathrm{2}}^{\mathrm{2}}\mathrm{-}{\mathit{n}}_{\mathrm{3}}^{\mathrm{2}}}{{\mathit{n}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{-}{\mathit{n}}_{\mathrm{3}}^{\mathrm{2}}}\left(5.1\right)$

Since *n*_{2} ≫ *n*_{3} and *n*_{1} ≥ *n*_{2}, the argument of the inverse tangent is much greater than unity but is not always large enough in the cases considered for arctan(*x*) to be approximated by π/2. Therefore, the simple mathematical identify arctan(*x*) = π/2—arctan(1/*x*) allows an accurate Taylor ...

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