Coupled Mode Theory

https://www.fiberoptics4sale.com/blogs/wave-optics/two-mode-coupling https://www.fiberoptics4sale.com/blogs/wave-optics/coupled-mode-theory http://home.iitj.ac.in/~k.r.hiremath/research/thesis.pdf http://home.iitj.ac.in/~k.r.hiremath/research/stoffer_hiremath_I_3.pdf

Coupling between different modes of the same waveguide

As previously, we assume, that the field is only propagating in \(x_3\)-direction. Not we include disturbances to the previously in \(x_3\)-direction translation-invariant system. As proposed in [1], we keep assuming that the mode field distributions are translation invariant and are eigenmodes of the undisturbed system, but include an \(x_3\)-dependence of the coefficients \(A_\nu(x_3)\).

\[ \begin{align}\begin{aligned} \mathcal{E}(\vec{x},x_3,t) = \sum_\nu A_\nu(x_3) \vec{E}_\nu(\vec{x})\mathrm{e}^{i\beta x_3}\\ \mathcal{H}(\vec{x},x_3,t) = \sum_\nu A_\nu(z) \vec{H}_\nu(\vec{x})\mathrm{e}^{i\beta x_3} \end{aligned}\end{align} \]

We use as previously Maxwell’s equations, but here we include a spatially dependen pertubation, which is represented by the additionally included polarization \(\mathcal{P}\). As we investigate a linear system, we assume that the pertubation at the same frequency \(\omega\)

\[ \begin{align}\begin{aligned} & \nabla\times\vec{\mathcal{E}} = i \mu_0 \omega \mathcal{H}\\ & \nabla\times\vec{\mathcal{H}} = -i \omega \varepsilon \mathcal{E} - i \omega \Delta \mathcal{P} \end{aligned}\end{align} \]

\[ \nabla \left( \mathcal{E}_1 \times \mathcal{H}_2^* + \mathcal{H}_2^* \times \mathcal{E}_1 \right) = - i \omega \left( \mathcal{E}_1 \cdot \Delta \mathcal{P}_2^* + \mathcal{E}_2^* \cdot \Delta \mathcal{P}_1 \right) \]

\[ \sum_\nu \frac{\mathrm{d}}{\mathrm{d}z} A_\nu(z) \mathrm{e}^{i(\beta_\nu-\beta_\mu)x_3} \int_\Omega \left( E_\nu \times H_\mu^* + E_\mu^* \times H_\nu \right) \cdot \hat{x_3} \mathrm{d}A = - i \omega \mathrm{e}^{-i\beta_\mu x_3} \int_\Omega E_\mu^* \cdot \nabla P \mathrm{d}A \]

\[ \frac{\mathrm{d} A_\nu(z)}{\mathrm{d}x_3} = - i \omega \mathrm{e}^{-i\beta_\nu x_3} \int_\Omega E_\nu^* \cdot \nabla P \mathrm{d}A \]

\[ \Delta \mathcal{P} = \Delta \varepsilon \mathcal{E} = \Delta \varepsilon \sum_\nu A_\nu(x_3) \vec{E}_\nu(\vec{x})\mathrm{e}^{i\beta_\nu x_3} \]

\[ \pm \frac{\mathrm{d} A_\nu(z)}{\mathrm{d}x_3} = \sum_\nu A_\nu(x_3) i \kappa_{\nu\mu} A_\mu \mathrm{e}^{i(\beta_\mu-\beta_\nu)x_3} \]

\[ \kappa_{\nu\mu} = \omega \int_\Omega E_\nu^* \cdot \Delta \varepsilon \cdot E_\mu P \mathrm{d}A \]

For lossles, i.e. \( \varepsilon = \varepsilon^* \)

\[ \kappa_{\nu\mu} = \kappa_{\mu\nu}^* \]

Bibliography

[1]

Jia-ming Liu. Photonic Devices. Cambridge University Press, Cambridge, edition, 2009. ISBN 978-1-139-44114-8.