## 5*Integrated Optics*

### 5.1 Planar Optical Waveguide Theory

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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