Coupled mode equations

For the further procedures, the unknown coefficients $C_{vi}$ are combined into amplitude vectors $\boldsymbol {C} = (\boldsymbol {C}_{\mbox{\scriptsize b}}, \boldsymbol {C}_{\mbox{\scriptsize s}}) = ((C_{\mbox{\scriptsize b}i}), (C_{\mbox{\scriptsize s}i}))$. To determine equations for these unknowns, here we follow an approach that relies on a variational principle [111,112]. Consider the functional

$$\begin{aligned}{\cal{F}(\boldsymbol {E}, \boldsymbol {H})} & = ... ...\boldsymbol {E}^{*} \right] \,\mbox{d}x\,\mbox{d}z, \end{aligned}$$

a 2-D restriction of the functional for the 3-D setting given in [43], stripped from the boundary terms. For the present 2-D configurations, the convention of vanishing derivatives $\partial_y = 0$ applies to all fields; the curl-operators are to be interpreted accordingly. $\cal{F}$ is meant to be viewed as being dependent on the six field components $\boldsymbol {E}$, $\boldsymbol {H}$. If $\cal{F}$ becomes stationary with respect to arbitrary variations of these arguments, then $\boldsymbol {E}$ and $\boldsymbol {H}$ satisfy the Maxwell curl equations as a necessary condition:

$$\nabla \times \boldsymbol {E} = - \omega \mu \boldsymbol {H}, \qquad \nabla \times \boldsymbol {H} = \omega \epsilon_{0} \epsilon \boldsymbol {E}.$$ (3.5)

We now restrict the functional to the fields allowed by the coupled mode ansatz. After inserting the trial field (3.3) into the functional (3.4), $\cal{F}$ becomes a functional that depends on the unknown amplitudes $\boldsymbol {C}$. For the ``best'' approximation to a solution of the problem (3.5) in the form of the field (3.3), the variation of ${\cal{F}}(\boldsymbol {C})$ is required to vanish for arbitrary variations $\delta\boldsymbol {C}$. Disregarding again boundary terms, the first variations of $\cal{F}$ at $\boldsymbol {C}$ in the directions $\delta C_{wj}$, for $j=1,\ldots,N_{w}$ and $w=$b,s, are

$$\delta {\cal{F}} = \int \sum_{v=\mbox{\scriptsize b,s}} \sum_{i=1... ...d}z} - {\sf {F}}_{vi, wj} C_{vi} \right \} \delta{C}_{wj}^{*}\,\mbox{d}z - c.c.$$ (3.6)

where $c.c.$ indicates the complex conjugate of the preceding integrated term,

$${\sf {M}}_{vi,wj} = \langle \boldsymbol {E}_{vi}^{}, \boldsymbol ... ...}_{wj}^{*} + \boldsymbol {E}_{wj}^{*} \times \boldsymbol {H}_{vi}^{} \right )\, x,$$ (3.7)

$${\sf {F}}_{vi,wj} = - \omega \epsilon_{0} \int (\epsilon - \epsilon_{v}) \boldsymbol {E}_{vi}^{} \cdot \boldsymbol {E}_{wj}^{*}\, x,$$ (3.8)

and where $\boldsymbol {a}_{z}$ is a unit vector in the $z$- direction. Consequently, one arrives at the coupled mode equations

$$\hspace{-0.1cm} \sum_{v=\mbox{\scriptsize b,s}} \sum_{i=1}^{N_{v}... ..._{v=\mbox{\scriptsize b,s}} \sum_{i=1}^{N_{v}} {\sf {F}}_{vi, wj}\, C_{vi} = 0,$$ (3.9)

for all $j=1,\ldots,N_{w}$ and $w=$b,s as a necessary condition for $\cal{F}$ to become stationary for arbitrary variations $\delta C_{wj}$. Note that the same expression is also obtained from the complex conjugate part of equation (3.6). In matrix notation, equations (3.9) read

$$(z)\, \frac{\mbox{d}\boldsymbol {C}(z)}{\mbox{d}z} = \mbox{\sf {F}}(z)\,\boldsymbol {C}(z).$$ (3.10)

Here the entries of the matrices M and F are given by the integrals (3.7) and (3.8). Due to the functional form of the bend modes and the varying distance between the bent and straight cores, these coefficients are $z$-dependent.