Etalon round-trip phase (${\varphi }_{\mathrm{rt}}$)

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.

An etalon coating group delay is the frequency derivative of the reflection phase contributed by the two coatings during one internal round trip. At fixed incidence and polarization, write the internal reflection coefficients as $r_j(\omega )=|r_j(\omega )|e^{i{\theta }_j(\omega )}$. With the phase convention of etalon round-trip phase,The phases must refer to consistently chosen surfaces: moving a reference plane transfers a propagation contribution between the terms. Only the consistently defined total is directly comparable with the resonance spacing.

A constant coating phase changes the positions of resonances but contributes zero group delay. A linear phase produces constant group delay. Curvature produces group delay dispersion:A highly reflecting coating therefore cannot be characterized by reflectance alone. The manufacturer's ultrafast-mirror documentation separately specifies reflection dispersion and shows multiple-bounce arrangements for compensation.^[ref]

For an original constant-slope example, suppose propagation contributes $10\mathrm{ps}$ and the coatings together contribute $0.20\mathrm{ps}$. The local free spectral range is approximately $1/(10.20\mathrm{ps})=98.04\mathrm{GHz}$, rather than $100\mathrm{GHz}$. Ignoring coating delay would overassign that $0.20\mathrm{ps}$ to propagation when inferring a spacer thickness or group index. The values are stipulated, not a coating specification.

For an independent curvature example, let the summed coating phase contain $D(\omega -{\omega }_0{)}^2/2$, with $D=100\mathrm{f}{\mathrm{s}}^2$. Across a frequency interval $10\mathrm{THz}$, its group delay changes by $D2\pi \Delta f\approx 6.28\mathrm{fs}$. The derivative is with respect to angular frequency; omitting $2\pi$ changes the result.

Experimentally, one may measure adjacent resonance frequencies and compare their spacing with an independently characterized spacer and group index. A discrepancy alone cannot uniquely identify coating delay: spacer dispersion, incidence angle and geometry also affect the result. Etalon delay inferred from free spectral range gives an interval-averaged phase slope; identifying it with a single-frequency derivative requires sufficiently slow variation. The simple manufacturer's spacer relation explicitly neglects spacer dispersion.^[ref]

Etalon reference-plane dependence is the change in the assigned propagation and reflection phases when the mathematical planes defining an etalon's field amplitudes are moved. A consistently computed etalon round-trip phase and its resonance frequencies remain unchanged. The decomposition into spacer delay and etalon coating group delay depends on those planes.

Use time dependence $e^{-i\omega t}$ and a lossless uniform spacer with wavenumber $k(\omega )=n(\omega )\omega /c$. Let the physical interfaces be separated by $L$, with internal reflection coefficients $r_1,r_2$ referred to their surfaces. The round-trip multiplier isThe multilayer formalism explicitly assigns forward and backward amplitudes at specified interfaces and propagates their phases through each layer. Its sign conventions must remain consistent when comparing reflection phases.^[ref]

Move the left mathematical reference plane a distance $\delta >0$ into the spacer, leaving the actual coating in place. A wave launched from that plane travels to the coating and back, so the effective left reflection coefficient becomes $r_1^{\prime }=r_1{e}^{2ik\delta }$. The explicit remaining distance is $L^{\prime }=L-\delta$. HenceThis original derivation changes the representation, not the physical cavity. Moving a real mirror would instead change the system.

Let ${\tau }_c=d\arg (r_1{r}_2)/d\omega$, using continuous wave phase. Since $dk/d\omega =n_g/c$, the new assigned terms obeyTheir sum equals ${\tau }_c+2n_{g}L/c$. If the spacer is dispersive, the corresponding second-derivative contributions likewise transfer between the two terms; the total group delay dispersion is unchanged.

For an original example, take $n_g=1.5$ and $\delta =15\mu \mathrm{m}$. The assigned reflection delay increases by about $150.1\mathrm{fs}$ while the assigned propagation delay decreases by the same amount. The free spectral range cannot reveal a change of bookkeeping. An apparent discrepancy of that size between two coating calculations can arise solely from different reference planes.

Report the incident medium, polarization, reference surfaces and phase convention with a coating-delay result. A measured resonance spectrum constrains the total phase; independently identifying a coating contribution requires a compatible spacer model. Etalon delay inferred from free spectral range therefore should not be compared with an isolated coating derivative until both calculations use the same reference planes.