Derivation of coupled mode equation by reciprocity technique

Alternatively, the coupled mode equations (3.9) or (3.10) can be derived by means of a ``reciprocity'' technique [43]. For any two electromagnetic fields $(\boldsymbol {E}_{p},\boldsymbol {H}_{p})$ and $(\boldsymbol {E}_{q}, \boldsymbol {H}_{q})$ with corresponding relative permittivity distributions $\epsilon_{p}$ and $\epsilon_{q}$, using the Maxwell equations one can derive the following identity,

$$\int \nabla \cdot \left(\boldsymbol {E}_{p} \times \boldsymbol {H}_{q}^{*} + \boldsymbol {E}_{q}^{*} \times \boldsymbol {H}_{p} \right )\, x= - \omega \epsilon_{0} \int (\epsilon_{p} - \epsilon_{q}) \boldsymbol {E}_{p} \cdot \boldsymbol {E}_{q}^{*}\, x,$$ (3.11)

commonly known as ``reciprocity identity'' or ``Lorentz reciprocity theorem''.

Apply Eq. (3.11) for $(\boldsymbol {E}, \boldsymbol {H}, \epsilon)$ and $(\boldsymbol {E}_{wj}, \boldsymbol {H}_{wj}, \epsilon_{w})$. After straightforward manipulations with the coupled mode field ansatz (3.3) leads to

$$\hspace{-3cm} \sum_{v=\mbox{\scriptsize b},\mbox{\scriptsize s}} ... ...ymbol {H}_{vi}) \ \mbox{d}x\ \frac{\mbox{d} C_{vi}}{\mbox{d}z} \hspace{0.5cm} +$$

$$\hspace{-4.5cm} \sum_{v=\mbox{\scriptsize b},\mbox{\scriptsize s}... ...*} + \boldsymbol {E}_{wj}^{*} \times \boldsymbol {H}_{vi}) \ \mbox{d}x \ C_{vi}$$

$$\hspace{4cm} = - \omega \epsilon_{0} \sum_{v=\mbox{\scriptsize b},\mbox{\scriptsiz... ...on_{w}) \boldsymbol {E}_{vi}\cdot \boldsymbol {E}_{wj}^{*} \ \mbox{d}x\ C_{vi}.$$ (3.12)

By applying the ``reciprocity identity'' for the second term on the left hand side of Eq.(3.12), and combining it with the right hand side, one obtains

$$\hspace{-3cm} \sum_{v=\mbox{\scriptsize b},\mbox{\scriptsize s}} ... ...dsymbol {H}_{vi}) \ \mbox{d}x\ \frac{\mbox{d} C_{vi}}{\mbox{d}z} \hspace{0.5cm}$$

$$\hspace{4cm} = - \omega \epsilon_{0} \sum_{v=\mbox{\scriptsize b},\mbox{\scriptsiz... ...on_{w}) \boldsymbol {E}_{vi}\cdot \boldsymbol {E}_{wj}^{*} \ \mbox{d}x\ C_{vi}.$$ (3.13)

Rewriting Eq. (3.13) in terms of the coefficients ${\sf {M}}_{vi, wj}$, ${\sf {F}}_{vi, wj}$ given by Eqs. (3.7), (3.8) leads to the coupled mode equations (3.9).