In this problem, we will study the eigenmodes of a resonator based on a fiber Bragg grating.

Let there be two Bragg gratings with length $l$ and grating strength $\kappa$ manufactured in an optical fiber at a distance of $2L$ from each other.

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.

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

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

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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.

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.

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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 $L \gg l$.

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.

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

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