``Standard model'' for resonators: Monomode setting

The ``standard resonator model'' [34,35] describes the frequency domain propagation of light inside the resonator. In this model, the optical field with vacuum wavelength $\lambda$ oscillates everywhere in time according to $\exp{(\mbox{i}\omega t)}$, where $\omega = k c$ is real angular frequency, $k=2 \pi / \lambda$ is the vacuum wavenumber, and $c$ is the vacuum speed of light. The model is based on the following approximations and assumptions:

• For simplicity, we here assume that all waveguides are monomodal and all modes are of same polarization.
• The waveguides are made of linear and nonmagnetic materials. The attenuation of the cavity fields is incorporated into the complex valued cavity mode propagation constant.
• The bent-straight waveguide couplers considered here are ``adiabatic''. Thus backreflections are negligible inside the couplers and the cavity segments.
• Outside the coupler regions, there is negligible ``interaction'' between the fields of the straight waveguides and the cavity.
• For further simplification, we assume that the resonator is symmetrical with respect to the $x$ and $z$ axes (see Figure 1.4).

Thus, the microresonator model under consideration consists of two identical bent-straight waveguide couplers (coupler I and coupler II) which are connected to each other by two segments of the cavity waveguide of length $L/2$. The external ports are constituted by straight waveguides. Variables $A, B, \tilde{A}, \tilde{B}$ (external connections) and $a, b, \tilde{a}, \tilde{b}$ (cavity connections) denote the amplitudes of properly normalized guided modes in the respective coupler port planes, which are identified by corresponding letters.

The response of the couplers is characterized by a scattering matrix S. For the coupler I and II, the relationship between coupler input and output amplitudes is given by

$$\begin{pmatrix}b \\ B \end{pmatrix} = \begin{pmatrix}a \\ A \end{pmatrix}, \hspace{0.5cm} \begin{pmatrix}\tilde{b} \\ \tilde{B} \end{pmatrix} = \begin{pmatrix}\tilde{a} \\ \tilde{A} \end{pmatrix}, = \begin{pmatrix}{\sf {S}}_{\mbox{\scriptsize bb}} & {\sf {S}}_{\... ...{S}}_{\mbox{\scriptsize sb}} & {\sf {S}}_{\mbox{\scriptsize ss}} \end{pmatrix}.$$ (1.1)

The entry ${\sf {S}}_{v,w}$ with $v, w =$   b$,$   s represents the ``coupling'' from the mode of waveguide $w$ to the mode of waveguide $v$. Thus ${\sf {S}}_{\mbox{\scriptsize bb}}$, ${\sf {S}}_{\mbox{\scriptsize ss}}$ are ``self coupling coefficients'', and ${\sf {S}}_{\mbox{\scriptsize bs}}$, ${\sf {S}}_{\mbox{\scriptsize sb}}$ are ``cross coupling coefficients''. Note that, here the matrix S is not assumed to be unitary. Losses, e. g.due to power transfer to radiative (non-guided) parts of the optical field in the coupler region, can thus be incorporated in S.

The (lossy) mode of the cavity waveguide is characterized by a complex valued cavity mode propagation constant $\gamma = \beta -$   i$\alpha$, where $\beta$ is the phase constant and $\alpha$ is the attenuation constant. Then for the propagation of the fields along the cavity segments, one writes

$$a = {\sf {G}} \tilde{b}, \hspace{0.5cm} \tilde{a} = {\sf {G}} b, \hspace{0.5cm} {\sf {G}} = ^{\displaystyle -\mbox{i}\beta L/2}~\mbox{e}^{\displaystyle - \alpha L/2}.$$ (1.2)