Within the framework of the theory of coupled modes, we will find solutions $a_{1,2}(x)$, $b_{1,2}(x)$ that do not require an external source: $a_1(-L-l)=0$, $b_2(L+l)=0$, and can have any amplitude.
The refractive index of the optical fiber is $n$.
A1
Write down the relationship between $a_1(-L)$ and $a_2(L)$, $b_1(-L)$ and $b_2(L)$ based on considerations of phase shift when a wave travels along an optical fiber.
The answer may contain $L$, $n$, and the wave number $k=\omega/c$, where $\omega$ is the frequency of light and $c$ is the speed of light.
The resulting equations can be reduced to the system
\[
\begin{cases}
a_2(L) + A b_1(-L) = 0 \\
B a_2(L) + b_1(-L) = 0,
\end{cases}\]
which has non-trivial solutions (i.e., solutions with any amplitude) when the self-consistency condition $1-AB=0$ is satisfied.
The resulting self-consistency condition cannot be satisfied exactly if $\kappa l$ is a finite number. This is because, in this case, the modes has a certain lifetime that can be found. To do this, we take into account that the amplitudes of all waves in the resonator decay as $e^{-\omega''t}$. For simplicity, we will assume that $L \gg l$.