Bent-straight waveguide couplers

Consider the coupler configuration shown in Figure 3.1(a). The coupled mode theory description starts with the specification of the basis fields, here the time-harmonic modal solutions associated with the isolated bent (b) and straight cores (c). Customarily, the real, positive frequency $\omega$ is given by the vacuum wavelength $\lambda$; we omit the common time dependence $\sim \exp{(\mbox{i}\omega t)}$ for the sake of brevity. Only forward propagating modes are considered, where, for convenience, we choose the $z$-axis of the Cartesian system as introduced in Figure 3.1 as the common propagation coordinate for all fields.

Figure: The bent-straight waveguide coupler setting (a). One assumes that the interaction between the waves supported by the bent and straight cores is restricted to the rectangular computational window $[x_{\mbox{\scriptsize l}}, x_{\mbox{\scriptsize r}}] \times [ z_{\mbox{\scriptsize i}}, z_{\mbox{\scriptsize o}}]$. Inside this region the optical field is represented as a linear combination of the modal fields of the bent waveguide (b) and of the straight waveguide (c).

Let $\boldsymbol {E}_{\mbox{\scriptsize b}p}$, $\boldsymbol {H}_{\mbox{\scriptsize b}p}$, and $\epsilon_{\mbox{\scriptsize b}}$ represent the modal electric fields, magnetic fields, and the spatial distribution of the relative permittivity of the bent waveguide. Due to the rotational symmetry, these fields are naturally given in the polar coordinate system $r$, $\theta$ associated with the bent waveguide. For the application in the CMT formalism, the polar coordinates are expressed in the Cartesian $x$-$z$-system, such that the basis fields for the cavity read

$$\begin{pmatrix}\boldsymbol {E}_{\mbox{\scriptsize b}p} \\ \boldsy... ...\ \boldsymbol {\tilde{H}}_{\mbox{\scriptsize b}p} \end{pmatrix}\!\!(r(x, z)) \, ^{\displaystyle -\mbox{i}\gamma_{\mbox{\scriptsize b}p} R \theta(x, z)}.$$ (3.1)

Here $\boldsymbol {\tilde{E}}_{\mbox{\scriptsize b}p}$ and $\boldsymbol {\tilde{H}}_{\mbox{\scriptsize b}p}$ are the radial dependent electric and magnetic parts of the mode profiles; the complex propagation constants $\gamma_{\mbox{\scriptsize b}p}$ prescribe the harmonic dependences on the angular coordinate. Note that the actual values of $\gamma_{\mbox{\scriptsize b}p}$ are related to the (arbitrary) definition of the bend radius $R$, see Section 2.4.1.

Likewise, $\boldsymbol {E}_{\mbox{\scriptsize s}q}$, $\boldsymbol {H}_{\mbox{\scriptsize s}q}$, and $\epsilon_{\mbox{\scriptsize s}}$ denote the modal fields and the relative permittivity associated with the straight waveguide. These are of the form

$$\begin{pmatrix}\boldsymbol {E}_{\mbox{\scriptsize s}q}\\ \boldsym... ...e s}q}\\ \boldsymbol {\tilde{H}}_{\mbox{\scriptsize s}q}\end{pmatrix}\!\!(x) \, ^{\displaystyle -\mbox{i}\beta_{\mbox{\scriptsize s}q} z},$$ (3.2)

i.e. guided modes with profiles $\boldsymbol {\tilde{E}}_{\mbox{\scriptsize s}q}$, $\boldsymbol {\tilde{H}}_{\mbox{\scriptsize s}q}$ that depend on the lateral coordinate $x$, multiplied by the appropriate harmonic dependence on the longitudinal coordinate $z$, with positive propagation constants $\beta_{\mbox{\scriptsize s}q}$. Note that for the present 2-D theory all modal solutions can be computed analytically. While the modal analysis is fairly standard for straight multilayer waveguides with piecewise constant permittivity, for the bend structures we employ analytic solutions in terms of Bessel- and Hankel functions of complex order, computed by means of a bend mode solver as presented in Chapter 2.

Now the total optical electromagnetic field $\boldsymbol {E}$, $\boldsymbol {H}$ inside the coupler region is assumed to be well represented by a linear combination of the modal basis fields (3.1), (3.2),

$$\begin{pmatrix}\boldsymbol {E}\\ \boldsymbol {H}\end{pmatrix}\!\!... ...begin{pmatrix}\boldsymbol {E}_{vi}\\ \boldsymbol {H}_{vi}\end{pmatrix}\!\!(x,z)$$ (3.3)

with so far unknown amplitudes $C_{vi}$ that are allowed to vary with the propagation coordinate $z$, and $N_{\mbox{\scriptsize b}}$, $N_{\mbox{\scriptsize s}}$ denote number of bent waveguide and straight waveguide modes under consideration. This assumption forms the central approximation of the present CMT approach; no further approximations or heuristics enter, apart from the numerical procedures used for the evaluation of the CMT equations (section 3.4). Note that here, unlike e.g. in Ref. [44], no ``phase matching'' arguments appear; via the transformation $r, \theta \rightarrow x, z$ the tilt of the wave front of the bend modes (3.1) is taken explicitly into account.