Consider a fiber where a Bragg grating of length $L$ is fabricated. According to the theory of coupled modes the amplitudes of the forward wave $a$ and backward wave $b$ are related by the following equations:
\[\begin{cases}
\dfrac{da}{dz} = i \kappa b \\[2pt]
\dfrac{db}{dz} = -i \kappa a
\end{cases},\]
where $\kappa$ is the strength of the grating.

A1 Reduce the system of two differential equations for two amplitudes $a$ and $b$ to a single equation for amplitude $a$.

Let's consider that the complex amplitude of the wave incident from the left is $A_0$.

A2 What boundary conditions are there for $a(0)$, $a(L)$, $a'(0)$ и $a'(L)$?

Remark: The existence of boundary conditions isn't necessary for all given values.

A3 Obtain the value of the reflection coefficient $R$.

A4 Does the energy loss occur due to the interaction of light with Bragg grating?