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.

A2  ^?? Express $a_1(-L)$ in terms of $b_1(-L)$. The answer may contain $\kappa$, $l$.

From the theory of coupled modes, we can obtain the equation for $a_1(x)$: $a_1'' - \varkappa^2 a_1 = 0$. Taking into account the boundary condition $a_1(-L-l)=0$, we have
\[b_1(-L)=ia_1(-L) \cdot \tanh \kappa L\]

A3  ^?? Express $b_2(L)$ in terms of $a_2(L)$. The answer may contain $\kappa$, $l$.

On the other side of the system, the same relationship is obtained through the reflection coefficient:
\[a_2(L)=ib_2(L) \cdot \tanh \kappa L\]

A4  ^?? Write down the condition for self-consistency when $\kappa l = \infty$. Find the frequencies of the resonator's eigenmodes $\omega_m$, assuming that $\kappa$ and $n$ do not depend on the wavelength.

System of equations:
\[
\begin{cases} a_2(L) - b_1(-L) \cdot ie^{2iknL} \coth \kappa L = 0\\
-a_2(L) \cdot ie^{2iknL} \coth \kappa L + b_1(-L) = 0
\end{cases}.\]
Self-consistency condition:
\[e^{4 ik nL} \coth^2 \kappa L = -1\]When $\kappa L = \infty$, we have the condition
\[4nL \frac{\omega_m}{c} = \pi + 2 \pi m\]

A5  ^?? Modify the equations found in question A1 to account for wave deacy during the propagation time from one grating to another.

From the new self-consistency condition, find the relationship between $\omega''$ and $\kappa l$ at $\kappa l \gg 1$ for each eigenmode of the resonator.

To account for wave decay, we need to add the factor $e^{-\omega''t}$, where $t$ is the time it takes for the wave to travel from one end of the resonator to the other. That is, the self-consistency condition will practically remain unchanged:
\[ e^{4 i knL} e^{-\omega'' 4nL/c} \coth^2 \kappa l = -1.\]From this it follows that
\[e^{-\omega'‘ 4nL/c} \simeq 1 - \frac{4nL \omega’'}{c}= \frac{\sinh^2 \kappa l}{\cosh^2 \kappa l}\simeq1-\frac{e^{-2\kappa l}}{4}\]

A6  ^?? Find the quality factors of the resonator's eigenmodes $Q_m$, assuming that $\kappa$ and $n$ do not depend on the wavelength.

The quality factor is determined by the ratio of the rate of decay of oscillations in the system to its frequency. For an oscillator with friction, the quality factor $Q$ is part of the equation of motion
\[ x'‘ + \frac{2\omega_0}{Q} x’ + \omega_0^2 x = 0,\]which has solutions
\[x = e^{-\omega_0/Q t} \left( A e^{i\omega_0 t} + Be^{-i \omega_0t} \right).
Therefore, the quality factor $m$ of the resonator mode is determined by the expression
\[Q_m = \frac{\omega_m}{\omega''}= 4\pi(1+2m)e^{2\kappa L} \]