deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 63125e3..6e2302a 100644  --- a/untitled.tex  +++ b/untitled.tex 
...
Many applications are concerned with the coupling between two modes. This coupling between two modes can be within the same waveguide or can be coupled between two parallel waveguides. For a system where we are interested in coupling two modes for either the parallel waveguides case or within the same waveguide, the two modes are described by two amplitudes A and B. The coupled equations are given by \[\pm \frac{ \partial A}{\partial z}=i \kappa_{aa} A+ i\kappa_{ab} B e^{i(\beta_b-\beta_a)z} \] \[\pm \frac{ \partial B}{\partial z}=i \kappa_{bb} B + i\kappa_{ba} A e^{i(\beta_a-\beta_b)z} \] for the two modes we want to solve. The coupling coefficients are given as part of a matrix $C=[C_{\nu \mu}]$ and $\widetilde{\kappa}=[\widetilde{\kappa}_{\nu \mu}]$ They are given as \[C_{\nu \mu}=\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \left( E_{\nu}^{*} \times H_{\mu} + E_{\mu} \times H_\nu \right) \cdot \hat{z} dxdy=c_{\mu \nu}^{*}\] \[ \widetilde{\kappa}_{\nu \mu}=\omega \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{\nu}^{*} \cdot \Delta \epsilon_{\mu} \cdot E_{\mu} dxdy\] where $\Delta \epsilon$ is the perturbation applied to the system. We then simplify the math further by removing the self coupling terms in equations (5) and (6) by expressing our normal mode coefficients by \[A(z)=\widetilde{A}(z)e^{\pm i \int_{0}^{z} \kappa_{aa}(z)dz} \] \[B(z)=\widetilde{B}(z)e^{\pm i \int_{0}^{z} \kappa_{bb}(z)dz}\]  For cases of interest, perturbation will either be independent of z or be a periodic funciton of z. This further reduces our coupling coefficients to \[ \pm \frac{\partial \widetilde{A}}{\partial z}=i\kappa_{ab} \widetilde{B}e^{i 2 \delta z} \] \[ \pm \frac{\partial \widetilde{B}}{\partial z}=i\kappa_{ba} \widetilde{A}e^{-i 2 \delta z}\]  The parameter of $2\delta$ is the phase mismatch between the two coupled modes.   \section{codirectional \section{Codirectional  Coupling}