If I have
$$\begin{aligned} \hat{\boldsymbol{a}}(z, \omega) &=e^{-i \frac{\omega}{c} z}\left(\zeta^{1 / 2} \hat{\boldsymbol{u}}(z, \omega)+\zeta^{-1 / 2} \hat{\boldsymbol{v}}(z, \omega)\right) \\ \hat{\boldsymbol{b}}(z, \boldsymbol{\omega}) &=e^{i \frac{\omega}{c} z}\left(-\zeta^{1 / 2} \hat{\boldsymbol{u}}(z, \omega)+\zeta^{-1 / 2} \hat{\boldsymbol{v}}(z, \omega)\right) \end{aligned}$$ with conditions $$\hat{\boldsymbol{a}}^{}(z=0, \omega)=\hat{\boldsymbol{f}}(\omega), \quad \hat{\boldsymbol{b}}^{}(z=L, \omega)=\left( \begin{array}{l}{0} \\ {0}\end{array}\right)$$
How do I show
$$\left\|\hat{\boldsymbol{a}}^{}(L, \omega)\right\|^{2}+\left\|\hat{\boldsymbol{b}}^{}(0, \omega)\right\|^{2}=\|\hat{\boldsymbol{f}}(\omega)\|^{2}$$
where $$\|\hat{\boldsymbol{a}}\|=\sqrt{\left|\hat{a}_{1}\right|^{2}+\left|\hat{a}_{2}\right|^{2}}$$ denotes the Euclidean norm?
($\zeta, c, L ,\omega$ are purely real.)