Transfer matrix and scattering matrix

To proceed further, the CMT equations are solved by numerical means. Brief details about the procedures are given in Section 3.4; the result can be stated in terms of a transfer matrix T that relates the CMT amplitudes at the output plane $z= z_{\mbox{\scriptsize o}}$ to the amplitudes at the input plane $z=z_{\mbox{\scriptsize i}}$ of the coupler region:

$$\boldsymbol {C}(z_{\mbox{\scriptsize o}}) = \mbox{\sf {T}}\,\boldsymbol {C}(z_{\mbox{\scriptsize i}}).$$ (3.15)

It remains to relate the transfer matrix, obtained directly as the solution of the CMT equations on the limited computational window, to the coupler scattering matrix as required for the abstract model of Section 1.4 or Section 4.1, respectively.

Outside the coupler (i.e. outside the region $[x_{\mbox{\scriptsize l}}, x_{\mbox{\scriptsize r}}] \times [ z_{\mbox{\scriptsize i}}, z_{\mbox{\scriptsize o}}]$), it is assumed that the interaction between the fields associated with the different cores is negligible. The individual modes propagate undisturbed according to the harmonic dependences on the respective propagation coordinates, such that the external fields are:

$$\begin{aligned}a_{p} \begin{pmatrix}\boldsymbol {\tilde{E}}_{\m... ...\quad \mbox{~for~} z \leq z_{\mbox{\scriptsize i}}, \end{aligned}$$

and

$$\begin{aligned}b_{p} \begin{pmatrix}\boldsymbol {\tilde{E}}_{\m... ...\quad \mbox{~for~} z \geq z_{\mbox{\scriptsize o}}. \end{aligned}$$

Here $\boldsymbol {a}=(a_{p})$, $\boldsymbol {A}=(A_{q})$ and $\boldsymbol {b}=(b_{p})$, $\boldsymbol {B}=(B_{q})$ are the constant external mode amplitudes at the input and output ports of the coupler (c.f. the corresponding definitions in the abstract resonator model of Section 1.4 or Section 4.1). See Figure 3.1 for the definitions of the coordinate offsets $z_{\mbox{\scriptsize i}}$, $\theta_{\mbox{\scriptsize i}}$ and $z_{\mbox{\scriptsize o}}$, $\theta_{\mbox{\scriptsize o}}$.

For a typical coupler configuration, the guided modal fields of the straight waveguide are well confined to the straight core. On the contrary, due to the radiative nature of the fields, the bend mode profiles can extend far beyond the outer interface of the bent waveguide. Depending upon the specific physical configuration, the extent of these radiative parts of the fields varies, such that also outside the actual coupler region, the field strength of the bend modes in the region close to the straight core may be significant. Therefore, to assign the external mode amplitudes $A_q$, $B_q$, it turns out to be necessary to project the coupled field on the straight waveguide modes.

At a sufficient distance from the cavity, in the region where only the straight waveguide is present, the total field $\phi = (\boldsymbol {E}, \boldsymbol {H})$ can be expanded into the complete set of modal solutions of the eigenvalue problem for the straight waveguide. The basis set consists of a finite number of guided modes $\phi_{\mbox{\scriptsize s}q} = (\boldsymbol {E}_{\mbox{\scriptsize s}q}, \boldsymbol {H}_{\mbox{\scriptsize s}q})$ and a nonguided, radiative part $\phi_{\mbox{\scriptsize rad}}$, such that

$$\phi = \sum_{q} B_{q} \phi_{\mbox{\scriptsize s}q} + \phi_{\mbox{\scriptsize rad}},$$ (3.18)

where $B_{q}$ are the constant amplitudes of $\phi_{\mbox{\scriptsize s}q}$. These amplitudes can be extracted by applying the formal expansion to the total field (3.3) as given by the solution of the CMT equations. Using the orthogonality properties of the basis elements, the projection at the output plane $z= z_{\mbox{\scriptsize o}}$ of the coupler yields

$$B_{q} \exp{(\mbox{i}\beta_{\mbox{\scriptsize s}q} z)} = C_{\mbox{... ...box{\scriptsize s}q}}{{\sf {M}}_{\mbox{\scriptsize s}q,\mbox{\scriptsize s}q}}.$$ (3.19)

where the mode overlaps $\langle \phi_{mi}; \phi_{nj} \rangle = \langle \boldsymbol {E}_{mi}^{}, \bolds... ... ; \boldsymbol {E}_{nj}^{}, \boldsymbol {H}_{nj}^{} \rangle = {\sf {M}}_{mi,nj}$ occur already in the coupled mode equations (3.10). An expression analogous to (3.19) can be written for the projection at $z=z_{\mbox{\scriptsize i}}$, where the coefficients $A_q$ are involved. What concerns the external amplitudes of the bend modes, no such procedure is required, since the field strength of the straight waveguide modes is usually negligible in the respective angular planes, where the major part of the bend mode profiles is located. Here merely factors are introduced that adjust the offsets of the angular coordinates in (3.16),(3.17) .

Thus, given the solution (3.15) of the coupled mode equations in the form of the transfer matrix T, the scattering matrix S that relates the amplitudes $a_p$, $b_p$, $A_q$, $B_q$ of the external fields as required in equation (1.1) is defined as

$$= \, \, ^{-1}$$ (3.20)

where P and Q are $(N_{\mbox{\scriptsize b}} + N_{\mbox{\scriptsize s}}) \times (N_{\mbox{\scriptsize b}} + N_{\mbox{\scriptsize s}})$ matrices with diagonal entries ${\sf {P}}_{p,p}=\exp{(-\mbox{i}\gamma_{\mbox{\scriptsize b}p} R \theta_{\mbox{\scriptsize i}})}$ and ${\sf {Q}}_{p,p} = \exp{(-\mbox{i}\gamma_{\mbox{\scriptsize b}p} R \theta_{\mbox{\scriptsize o}})}$, for $p = 1,\ldots,N_{\mbox{\scriptsize b}}$, followed by the entries ${\sf {P}}_{q+{N_{\mbox{\scriptsize b}}},q+{N_{\mbox{\scriptsize b}}}} = \exp{(-\mbox{i}\beta_{\mbox{\scriptsize s}q} z_{\mbox{\scriptsize i}})}$ and ${\sf {Q}}_{q+{N_{\mbox{\scriptsize b}}},q+{N_{\mbox{\scriptsize b}}}} = \exp{(-\mbox{i}\beta_{\mbox{\scriptsize s}q} z_{\mbox{\scriptsize o}})}$, for $q = 1,\ldots,N_{\mbox{\scriptsize s}}$.

A lower left block is filled with entries ${\sf {P}}_{q+{N_{\mbox{\scriptsize b}}},p} = \exp{(-\mbox{i}\beta_{\mbox{\scr... ...criptsize s}q,\mbox{\scriptsize s}q} \right \vert _{z=z_{\mbox{\scriptsize i}}}$ and ${\sf {Q}}_{q+{N_{\mbox{\scriptsize b}}},p} = \exp{(-\mbox{i}\beta_{\mbox{\scr... ...criptsize s}q,\mbox{\scriptsize s}q} \right \vert _{z=z_{\mbox{\scriptsize o}}}$, for $q = 1,\ldots,N_{\mbox{\scriptsize s}}$ and $p = 1,\ldots,N_{\mbox{\scriptsize b}}$, respectively, that incorporate the projections. All other coefficients of P and Q are zero.