PX284 - U3 - fabry-perot interferometry

phase difference: $2 ϕ = 2 d k_{2} \cos t$
transmitted wave:

t_{t o t a l} = t t^{'} (1 + r^{2} e^{i 2 ϕ} + (r^{2} e^{i 2 ϕ})^{2} + \dots)

this is a geometric progression:

t_{t o t a l} = \frac{t t^{'}}{1 - r^{2} e^{i 2 ϕ}}

transmitted power: $T = | t_{t o t a l} |^{2}$
assuming $t t^{'}$ and $r$ are real:

\begin{aligned} T & = \frac{t^{2} t^{' 2}}{(1 - r^{2} e^{i 2 ϕ}) (1 - r^{2} e^{- i 2 ϕ})} \\ = \frac{t^{2} t^{' 2}}{1 + r^{4} - 2 r^{2} \cos 2 ϕ} \\ = \frac{t^{2} t^{' 2}}{(1 - r^{2})^{2} 2 r^{2} (1 - 2 \sin^{2} ϕ)} \\ = \frac{t^{2} t^{' 2} / (1 - r^{2})^{2}}{1 + F \sin^{2} ϕ} \end{aligned}

where, $F = 2 r^{2} / (1 - r^{2})^{2}$ is called "finesse"

if no losses:

t^{2} = t^{' 2} = (1 - r^{2}) = 1 - R ⟹ T = \frac{1}{1 + F \sin^{2} ϕ}

F = \frac{4 r^{2}}{(1 - r^{2})^{2}} = \frac{4 R}{(1 - R)^{2}}

\begin{matrix} R = 0.5 ⟹ F = 8 \\ R = 0.9 ⟹ F = 360 \end{matrix}

fabry-perot interferometer acts as a narrow band filter
the value of $ϕ = - = k_{2} d \cos t$ can be tuned to pass specific wavelengths, $\frac{2 π}{λ_{0}} n_{2} d \cos t$