 Etalon round-trip phase

An etalon round-trip phase is the phase accumulated by a circulating optical field after one complete internal round trip, relative to its earlier passage. At normal incidence through a uniform plate of thickness $L$,where the last terms are the phases of the two internal amplitude-reflection coefficients. Propagation and reflection both contribute; a coating phase is not automatically zero.

The resonance condition follows from a field sum. If $r_1{r}_2{e}^{2ikL}=a{e}^{i{\varphi }_{\mathrm{rt}}}$, with $0\le a<1$, successive transmitted paths contain factors $1,a{e}^{i{\varphi }_{\mathrm{rt}}},a^2{e}^{2i{\varphi }_{\mathrm{rt}}},\dots$. Their geometric series has denominator $1-a{e}^{i{\varphi }_{\mathrm{rt}}}$. Its squared magnitude containsFor slowly varying amplitudes, transmission peaks occur near ${\varphi }_{\mathrm{rt}}=2\pi m$. The primary manufacturer tutorial describes the resulting etalon transmission peaks and distinguishes their spacing from their width.^[ref]

A constant phase offset shifts resonance positions without changing the local slope. In an original model with constant round-trip delay $20\mathrm{ps}$, adding reflection phase $0.4\mathrm{rad}$ shifts a given resonance by approximately $-0.4/(2\pi \times 20\mathrm{ps})=-3.18\mathrm{GHz}$. The free spectral range remains $50\mathrm{GHz}$ in this constant-slope model.

Frequency-dependent phase has a different effect. Differentiating with respect to angular frequency givesThe last two terms form an etalon coating group delay. A stipulated $20\mathrm{ps}$ propagation contribution plus $2\mathrm{ps}$ coating contribution gives local resonance spacing about $1/(22\mathrm{ps})=45.45\mathrm{GHz}$, rather than 50 GHz. These are illustrative phase functions, not measured coating specifications.

Use a continuous branch of wave phase when differentiating: jumps of $2\pi$ in a wrapped display are not physical delay spikes. The etalon delay inferred from free spectral range article derives the interval relation between phase slope and adjacent resonances. Finite pulse bandwidth, angular spread and frequency-dependent reflection amplitudes determine how closely the simple resonance model describes an actual optical pulse.