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\section{Introduction}  Coupled-mode theory is concerned with coupling spatial modes of differing polarizations, distribution, or both. To understand codirectional coupling it is useful to have an understand of background material that builds to codirectional coupling. First, consider coupling normal modes in a single waveguide that is affected by a perturbation. Such case is single-waveguide mode coupling. The perturbation in question is spatially dependent and is represented as $\Delta P(r)$, a perturbing polarization. Consider the following Maxwell's equations \[ \nabla \times E=i \omega \mu_0 H \\ \nabla \times H=-i \omega \epsilon E-i \omega \Delta P\] Consider two sets of fields $(E_1,H_1)$ and $(E_2,H_2)$, they satisfy the Lorentz reciprocity theorem give by $\nabla \cdot \left( E_1 \times H_2^{*} +E_2^{*} \times H_1\right)=- i \omega \left( E_1 \cdot \Delta P_2^{*}-E_2^{*} \cdot \Delta P_1 \right)$   For $\Delta P_1=\Delta P$ and $\Delta P_2=0$ and integrating over the result for the cross section of the waveguide in question, we get \[ \sum_{\nu} \frac{ \partial }{\partial z} A_{\nu}(z)e^{i\left(\beta_{\nu}-\beta_{\mu}\right)} z=i \omega e^{-i \beta_{\nu} z} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{\mu}^{*} \cdot \Delta P dxdy \] Evoking orthonormality, we can get the coupled-mode equation \[ \pm \frac{ \partial A_{\nu}  }{\partial z} A_{\nu}=i =i  \omega e^{-i \beta_{\mu} z} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{\nu}^{*} \cdot \Delta P dxdy \]