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For identical mirrors with power reflectivity $\mathcal{R}$, transmission $\mathcal{T}$ and absorption $1-\mathcal{R}-\mathcal{T}$, summing round trips at normal incidence givesHere $\delta$ includes propagation and both internal reflection phases. Optical absorption reduces the resonant peak below unity, while a nonzero reflection phase shifts the resonance condition $\delta =2\pi m$. Frequency-dependent mirror phase and refractive index can also change the free spectral range.

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]

An etalon delay inferred from free spectral range estimates an etalon round-trip delay from the frequency spacing of adjacent transmission resonances. When round-trip group delay varies little across that spacing,The measured quantity is resonance spacing, not the width of one resonance. The manufacturer tutorial explicitly distinguishes free spectral range from optical finesse, which compares that spacing with an individual peak's width.^[ref]

The relation follows from wave phase. Let $\varphi (\nu )$ be the total etalon round-trip phase, including relevant reflection phases. Adjacent resonances satisfy $\varphi ({\nu }_{m+1})-\varphi ({\nu }_m)=2\pi$. Defining ${\tau }_{\mathrm{rt}}(\nu )=(2\pi {)}^{-1}d\varphi /d\nu$ gives the exact interval relationThus the reciprocal spacing is the delay averaged over that frequency interval. Interpreting it as a local pulse-envelope delay requires sufficiently weak variation across the interval and relevant pulse spectrum.

For a uniform plate at normal incidence with negligible frequency-dependent reflection phase, $\varphi =4\pi \nu n(\nu )L/c$. Differentiation gives ${\tau }_{\mathrm{rt}}=2n_{g}L/c$, where $n_g=n+\nu dn/d\nu$ is the group refractive index. Treating $n$ as frequency independent recovers the manufacturer tutorial's elementary plate formula; using a phase index in a strongly dispersive material can instead give the wrong delay.

An original measured-spacing example takes $\Delta {\nu }_{\mathrm{FSR}}=25.0\mathrm{GHz}$. The inferred delay is $40.0\mathrm{ps}$. A spacing standard uncertainty of $0.2\mathrm{GHz}$ contributes approximately $0.32\mathrm{ps}$ to the delay uncertainty through $u(\tau )=u(\Delta \nu )/(\Delta \nu {)}^2$. With stipulated $n_g=1.50$ and negligible coating delay, the corresponding plate thickness is about $4.00\mathrm{mm}$.

In a spectral experiment, identify adjacent peaks of the same resonance family and calibrate the frequency coordinate. If the coordinate is wavelength, use $\Delta \nu =c|1/{\lambda }_1-1/{\lambda }_2|$ rather than treating a wavelength interval as a frequency interval. Beam angle, dispersion and reflection phase belong in the propagation model.

A time-domain check looks for replicas separated by the predicted extra round trip. The historical single-shot correlator experiment inserted a known-thickness BK7 etalon to calibrate delay and separately placed an uncoated etalon in the input to create a double-reflection postpulse.^[ref] The single-pass delay caused by inserting glass into an arm is distinct from the additional round-trip separation of successive transmitted replicas.

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.

For an ideal air-gap Fabry–Pérot etalon at normal incidence, the free spectral range is ${\nu }_{\mathrm{FSR}}=c/(2d)$ and the peak full width at half maximum is $\delta \nu ={\nu }_{\mathrm{FSR}}/\mathcal{F}$. To separate a pair of lines spaced by $\Delta \nu$ by at least $M$ peak widths, require $d\mathcal{F}\ge Mc/(2\Delta \nu )$. Putting both lines inside one unambiguous free spectral range additionally requires $d<c/(2\Delta \nu )$, so $\mathcal{F}>M$. In terms of their vacuum wavelengths, $c/\Delta \nu ={\lambda }_1{\lambda }_2/|{\lambda }_2-{\lambda }_1|$.

The strict order condition needs a margin if tails from adjacent periodic peaks are to be negligible. If every neighbouring peak in a scanned two-line periodic spectrum must be separated by $M$ widths, the stronger condition is $\min (\Delta \nu ,{\nu }_{\mathrm{FSR}}-\Delta \nu )\ge M{\nu }_{\mathrm{FSR}}/\mathcal{F}$. Ordinary resolving power alone does not prevent ambiguous repeated orders. A fixed etalon's transmission as a function of wavelength is a periodic-in-frequency filter; it is not automatically a plot of the incident source's spectral lines.

An etalon round-trip delay is the extra envelope travel time associated with one additional internal round trip in an etalon. At normal incidence, for a transparent plate of thickness $L$ and a sufficiently narrowband optical pulse,The relevant quantity is group refractive index, since the delay concerns the pulse envelope. The extra propagation phase is $\varphi (\omega )=2\omega n(\omega )L/c$; its derivative gives $d\varphi /d\omega ={\tau }_{\mathrm{rt}}$. This assumes negligible frequency-dependent reflection phase and treats dispersion locally.

For an original chosen plate with $L=3.0\mathrm{mm}$ and $n_g=1.60$, the delay is about $32.0\mathrm{ps}$. A pulse transmitted after one extra round trip appears that much later than the directly transmitted pulse. Another extra round trip adds the same delay in this local model, producing an echo sequence. The optical postpulse article separately describes the reflection losses that determine echo energies.

The frequency-domain connection gives an independent consistency check. Neighboring transmission resonances differ in round-trip phase by $2\pi$, so the local free spectral range satisfiesThe example predicts about $31.2\mathrm{GHz}$. This etalon delay inferred from free spectral range uses the same local dispersion assumptions; the spacing can vary across a broad spectrum.

The historical third-order correlator calibrated its delay coordinate by inserting a BK7 etalon of known thickness into an arm and measuring the trace shift.^[ref] A plate inserted into an arm also changes the direct propagation delay relative to the displaced air path; that single-pass insertion delay is distinct from the echo's extra internal round trip. Oblique incidence, dispersive coatings and broad-band pulse reshaping require a more complete propagation model.

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.

At normal incidence, with no optical dispersion and zero round-trip reflection wave phase modulo $2\pi$, the resonance condition in Fabry–Pérot transmission from multiple reflections isHere $\nu$ is frequency and $c$ is the speed of light in vacuum. A change of $\nu$ by one free spectral range advances the round-trip wave phase by $2\pi$. At fixed internal angle of incidence $\theta$, the same nondispersive model gives $c/(2nd\cos \theta )$. Holding the external angle of incidence fixed in a dispersive medium need not hold the internal $\theta$ fixed, because of Snell's law.

For stipulated $n=1$ and $d=10\mathrm{mm}$, the free spectral range is $14.9896229\mathrm{GHz}$ using $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$. A frequency difference of $0.10\Delta {\nu }_{\mathrm{FSR}}$ is therefore $1.49896229\mathrm{GHz}$. Transmission repeats every free spectral range: one isolated scan cannot distinguish that separation from $1.10\Delta {\nu }_{\mathrm{FSR}}$. Independent knowledge of the optical spectrum, an additional filter, or a changed mirror separation is needed to assign orders.

Equal spacing in frequency does not imply equal spacing in wavelength. At normal incidence the exact adjacent resonance wavelengths obeyFor example, choosing $2nd=10{\lambda }_{10}$ makes the exact gap to order 11 equal to ${\lambda }_{10}/11$, while the high-order estimate gives ${\lambda }_{10}/10$. The approximation is already ten percent larger at order 10. With optical dispersion or frequency-dependent reflection wave phase, obtain local spacings from the total phase difference instead: $\Delta \nu \simeq 2\pi /(d\delta /d\nu )$ when its derivative varies little over one order.

The finesse of a Fabry–Pérot interferometer is its free spectral range divided by a transmission peak's full width at half maximum. Large finesse means a narrow accepted frequency interval compared with the spacing between repeated orders. Here half maximum means $T=1/2$ of the ideal unit peak, with zero detector background subtracted; it does not mean halfway between the transmission minimum and maximum.

Write $A=4R/(1-R{)}^2$ for the coefficient multiplying ${\sin }^2(\delta /2)$ in Fabry–Pérot transmission from multiple reflections. This coefficient is sometimes called the coefficient of finesse; it is not the finesse $\mathcal{F}$. About a resonance $\delta =2\pi m$, put $ϵ=\delta -2\pi m$. Solving $T=1/2$ givesand, since one free spectral range advances the wave phase by $2\pi$,These are exact for the ideal nondispersive model with constant $R$ and linear frequency dependence of $\delta$. The inverse sine requires $(1-R)/(2\sqrt{R})\le 1$, or $R\ge 3-2\sqrt{2}\simeq 0.171573$. At equality, neighbouring peaks merely touch half maximum at their shared minima: the nominal width is one full free spectral range and $\mathcal{F}=1$. Isolated half-maximum peaks require the strict inequality. Below this threshold the transmission never falls to $1/2$, so this peak-width definition supplies no finesse; at $R=0$ the transmission is flat.

For large reflectivity, the small-angle approximation $\arcsin x\simeq x$ givesThis is a high-reflectivity approximation, not an identity for every $R$. The original dimensionless examples below have exact full width at half maximum values obtained from the inverse sine formula.

The beauty of the narrowing is visible directly in the geometric series: large $R$ preserves contributions from many round trips, so a small phase difference accumulates across many terms and spoils their addition. For the stipulated $10\mathrm{mm}$ vacuum gap, $R=0.95$ gives an ideal full width at half maximum of about $244.792\mathrm{MHz}$. This instrumental width is not a measured spectral line width. A real scan also depends on the source profile, accepted angles, mirror shape, alignment, and scan stability. The University of Washington handout separately discusses mirror-reflectivity, mirror-flatness, and detector-pinhole contributions; increasing $R$ alone cannot remove the others.^[ref]

The free spectral range is the wavelength or frequency spacing between adjacent orders or resonances of an optical instrument. It identifies when different orders can overlap. Always specify the coordinate: a uniform spacing in frequency need not be uniform in wavelength. For an ideal Fabry–Pérot interferometer at normal incidence, with mirror separation $d$ and constant refractive index $n$, Fabry–Pérot resonance spacing gives $\Delta {\nu }_{\mathrm{FSR}}=c/(2nd)$, where $c$ is the speed of light in vacuum. The free spectral range of an echelle grating follows a different order condition.

At fixed incident and central outgoing angles, the grating equation gives $m{\lambda }_m=K=d(\sin \alpha +\sin {\beta }_0)$. Adjacent central wavelengths are separated byThis is the usual adjacent-order free spectral range. A detector width of approximately $q\Delta {\lambda }_{\mathrm{FSR}}$, where $q$ is the grating dispersion, covers one such interval.

The closest components above differ by $\Delta {\nu }_{\min }={\mu }_{B}B/(2h)\simeq 7.0\mathrm{GHz}{\mathrm{T}}^{-1}B$. At $B=0.1\mathrm{T}$, their separation is about $0.70\mathrm{GHz}$. A visible spectral line near $500\mathrm{nm}$ would consequently require spectral resolving power of order $\nu /\Delta {\nu }_{\min }\simeq 9\times 10^5$. An ordinary low-resolution spectrometer would merge the components.

A suitable arrangement places an excited low-pressure strontium discharge or a transversely observed atomic beam between wide electromagnet pole pieces. Collect the emission perpendicular to the magnetic field, select the desired spectral line with a prefilter or monochromator, and send it through a high-finesse scanning Fabry–Pérot interferometer to a detector. A rotatable linear polarizer separates the two $\pi$ components from the four $\sigma$ components. Calibrate both the interferometer's free spectral range and the local magnetic field; take zero-field and several field-strength scans, and fit component positions against $B$.

The magnetic field should be uniform across the viewed volume, with its associated Zeeman splitting larger than the combined natural linewidth, Doppler broadening, pressure broadening and instrumental width, but much smaller than the separations of the relevant atomic fine-structure levels. A collimated transverse atomic beam, low collision rate and restricted collection angle reduce unwanted spectral line broadening. Avoid appreciable power broadening, resolve or model isotope and hyperfine structure, and check for self-absorption. A zero-field scan and the predicted linear scaling help distinguish actual Zeeman splitting components from neighbouring spectral lines.

Let $\mathcal{R}=|r{|}^2$ be the single-mirror power reflectivity and ${\theta }_g$ the propagation angle inside the gas. Define $\lambda$ as the vacuum wavelength. Successive transmitted complex amplitudes differ by a round-trip factor $\mathcal{R}{e}^{i\delta }$, where, for real dielectric reflection coefficients whose product has positive sign,The cosine arises by comparing successive emerging rays on the same outgoing wavefront; it is not the literal zigzag distance $2d/\cos {\theta }_g$. Equivalently, the round-trip phase is $2k_{z}d$. If the source's $\theta$ is the angle in external air, Snell's law gives $n\sin {\theta }_g=\sin \theta$ andIf $\theta$ denotes the internal angle, simply set ${\theta }_g=\theta$.

For identical lossless mirrors, their power transmission is $1-\mathcal{R}$. In an energy-normalized amplitude convention this is $|t{|}^2=1-\mathcal{R}$; more general interface amplitudes require the corresponding wave impedance factors. The geometric series of transmitted amplitudes givesHenceThis is Fabry–Pérot transmission from multiple reflections.^[ref]

At normal incidence, for nondispersive $n$ and zero net reflection phase modulo $2\pi$, the resonance condition is $\delta =2\pi m$. The resonance frequencies and free spectral range areFor finite $\mathcal{R}<1$ the resonances have finite width, and transmission away from them is nonzero. Thus “only discrete frequencies” is an ideal high-reflectivity description of the selected resonances, not an exact statement about the support of the transmitted spectrum. For example, $\mathcal{R}=0.9$ at $\delta =\pi$ still gives $I_t/I_0=(0.1/1.9{)}^2\simeq 0.00277$.

At half the unit peak, $Q{\sin }^2(\delta /2)=1$. Measuring the phase displacement from a resonance yieldsThe last expression is the high-reflectivity approximation, with finesse $\pi \sqrt{\mathcal{R}}/(1-\mathcal{R})$. At $\mathcal{R}=0.9$, the full phase width is about $0.2109$ radians and $\Delta {\nu }_{\mathrm{FWHM}}/\Delta {\nu }_{\mathrm{FSR}}\simeq 0.03357$. At $\mathcal{R}=0.1$, the minimum transmission is $0.6694$, so the trace never reaches half the peak and this full width at half maximum is not defined. The exact half-maximum condition requires $\mathcal{R}\ge 3-2\sqrt{2}$.

For the absorbing Fabry–Pérot mirror phase and transmission, write $r=\sqrt{\mathcal{R}}{e}^{i{\varphi }_r}$ and let $\mathcal{T}$ be the power transmission of one mirror. Optical absorption makes $\mathcal{R}+\mathcal{T}<1$. The transmitted intensity becomesHere the phase formula is at normal incidence. The peak is reduced below unity, while a change in $\mathcal{R}$ changes the width. The complex reflection phase also moves the resonances: if ${\varphi }_r$ is approximately constant,If either $n$ or ${\varphi }_r$ depends on frequency, this is an implicit condition and the free spectral range need not be constant. No numerical shift is determined without the mirrors' optical response.

A longitudinal mode makes an integer round-trip phase accumulation. For the same transverse mode and an empty nondispersive cavity, increasing that integer by one changes the optical frequency by the free spectral rangeFrequency-dependent mirror phases or a dispersive intracavity medium would change the round-trip group delay used here.

For a paraxial ray, use the height-slope column $(x,\theta {)}^T$. Free propagation and reflection from a concave mirror are represented, in unfolded coordinates, byThe reflection changes the slope by $-2x/R$. Because the mirrors are identical, the same half-round-trip matrix is applied repeatedly:Its two eigenvalues are reciprocal. They lie on the unit circle without a growing component when $|1-l/R|<1$, whereas $l>2R$ gives a real eigenvalue of modulus greater than one. At $l=2R$, the repeated eigenvalue has a nontrivial Jordan block and generic rays drift out. Thus the nondegenerate stable interval isThe confocal value $l=R$ is included: there $A^2=-I$, so the repeated round trip is bounded. Applying a strict trace inequality to the full-round-trip matrix without checking this exact matrix would incorrectly discard that point. These are the paraxial conditions for a symmetric concave-mirror resonator.

Symmetry places the Gaussian beam waist halfway between the mirrors. Its Rayleigh range is $z_R=\pi w_0^2/\lambda$, and its wavefront radius at distance $z$ from the waist is $R_b(z)=z+z_R^2/z$. Matching that wavefront to the mirror at $z=l/2$ givesHenceA real positive Rayleigh range requires the same stable interval. At its limiting geometries the finite confined Gaussian mode degenerates; the formula is not a way to realize an arbitrarily small physical waist without leaving the paraxial model.

Two parallel partially reflecting surfaces in a Fabry–Pérot interferometer produce a sequence of coherent transmitted beams. For equal lossless mirror intensity reflectance $\mathcal{R}$ and round-trip phase $\delta =4\pi nd\cos {\theta }_t/\lambda$, summing the geometric series of amplitudes givesresonances have $2nd\cos {\theta }_t=m\lambda$ and unit peak transmission in this symmetric lossless model. Varying angle gives concentric equal-inclination rings; changing spacing scans the resonance wavelengths.

At normal incidence with negligible dispersion, the free spectral range is $\Delta {\nu }_{\mathrm{FSR}}=c/(2nd)$. For high reflectance, the peak full width gives finesse and resolving powerThe coefficient $\mathcal{F}$ in the transmission formula and the finesse $\mathcal{F}$ are different quantities. Multiple round trips sharpen a peak, allowing high-resolution spectroscopy, frequency selection and cavity measurements. Absorption, mirror mismatch, finite angular spread, imperfect parallelism and surface errors reduce usable transmission or finesse. Adjacent orders can overlap in a broadband measurement, requiring order selection or prior spectral information.

Let the first transmitted complex amplitude be $T{E}_{\mathrm{in}}{e}^{ikd}$: two amplitude transmission factors each of magnitude $\sqrt{T}$ give magnitude $T$. Each extra internal round trip adds two amplitude reflections, with combined magnitude $R$, and phase $\delta$. With the mirror phase convention already incorporated in the stipulated $\delta$,Taking the squared modulus givesIt is $R$, not $R^2$, that multiplies the field after a round trip; $R^2$ would be the corresponding round-trip intensity fraction.

At resonance $\delta =2m\pi$, $B_{\max }=T^2/(1-R{)}^2$. If $h$ is the positive phase displacement to half maximum,For $R=0.80$, the stipulated high-reflectivity approximation gives $h=0.2236\mathrm{rad}$, while the exact value is $0.2241\mathrm{rad}$. The full width is twice this value. With $T=0.20$, $B_{\max }=1$, $B_{\min }=T^2/(1+R{)}^2=1/81$ and the pattern repeats every $2\pi$. Its ideal finesse is approximately $\pi \sqrt{R}/(1-R)=14.05$.

A wavelength component resonates at $d=m\lambda /2$. Within the scan interval, the individual resonances areNo neighboring orders of these lines fall within the stated interval. The observed signal is the sum of their two Airy transmissions, since different spectral components add intensities. Individual resonance positions identify the two peaks to the plotted precision; the tails cause very small shifts of the maxima of the sum.

The gap-coordinate full width isThe separation is $0.100\mu \mathrm{m}$, about $2.8$ full widths, so the two components can be resolved by the ideal etalon. The equivalent wavelength resolution near order $1000$ is about $\lambda /(m\mathcal{F})\simeq 0.071\mathrm{nm}$, smaller than their $0.2\mathrm{nm}$ separation.

The added wavelength has an order-$999$ resonance atIt lies between the preceding two peaks; its adjacent orders are outside the scan. Its input strength was not specified, so the third peak's height cannot be uniquely assigned. The original plot uses equal strength only as a declared illustration.

This is an order-identification problem: the free spectral range near $d=0.5\mathrm{mm}$ is $\Delta {\lambda }_{\mathrm{FSR}}\simeq {\lambda }^2/(2d)\simeq 1\mathrm{nm}$. A line more than one free spectral range away can appear among peaks of another order. A lower-resolution prefilter or additional dispersive spectrometer is needed to identify the order; a narrow peak alone does not identify an absolute wavelength. mirror parallelism, beam collimation and detector angular acceptance must also be controlled before attributing a measured width entirely to reflectivity.^[ref]

Let ${\lambda }_1=589.0\mathrm{nm}$, ${\lambda }_2=589.6\mathrm{nm}$ and $\Delta \lambda =0.6\mathrm{nm}$. Requiring the separation to be at least $100$ times the measured full width at half maximum means a width no larger than $0.006\mathrm{nm}$, or a resolving power of approximately $589.3/0.006\simeq 9.8\times 10^4$.

The etalon resolving power and order overlap constraints are particularly clear in frequency. Put $\Delta \nu =c|1/{\lambda }_1-1/{\lambda }_2|$. At normal incidence, the free spectral range is ${\nu }_{\mathrm{FSR}}=c/(2d)$ and the instrumental width is $\delta \nu ={\nu }_{\mathrm{FSR}}/\mathcal{F}$. ThusAvoiding ambiguity between the doublet and adjacent interference orders requires a free spectral range exceeding the doublet separation:Together these giveUsing the narrow-peak finesse from the previous part, $\pi \sqrt{R}/(1-R)>100$ gives $\boxed{R\gtrsim 0.9691}$, with a practical margin above that limiting value. At the exact limit, the gap between orders vanishes, so equality is not a sensible operating point.

One feasible ideal design is $R=0.99$, $\mathcal{F}\simeq 313$, and $d=0.14018\mathrm{mm}$. It has an approximate wavelength free spectral range of $1.24\mathrm{nm}$ and instrumental width $0.0040\mathrm{nm}$ near the doublet. The doublet spacing is about $151$ widths. Even the spacing to the nearest replica of the other line exceeds $100$ widths in this example. If that latter separation is also required in general, impose the stronger condition $\min (\Delta \nu ,{\nu }_{\mathrm{FSR}}-\Delta \nu )\ge 100{\nu }_{\mathrm{FSR}}/\mathcal{F}$, rather than just the two basic inequalities.

Such mirror reflectivities and submillimetre gaps are plausible for a high-quality Fabry–Pérot etalon. Flatness, parallelism, coating loss, angular spread, mechanical stability and source width limit the achieved finesse. A wavelength width of $0.004\mathrm{nm}$ corresponds to a gap variation of order $d\delta \lambda /\lambda \simeq 1\mathrm{nm}$, so the ideal coating-limited estimate demands stringent uniformity. The paper expressly excludes intrinsic source broadening.

The first panel is the fixed etalon's transmission versus wavelength, a comb of resonances. It does not by itself have one peak at each sodium line. The second panel shows the two peaks obtained by scanning the etalon phase or gap and calibrating the scan to wavelength. A fixed-gap output spectrum is instead the incident spectrum multiplied by the transmission function. This distinction is needed to interpret a doublet measurement correctly.

A high-finesse Fabry–Pérot etalon obtains very narrow transmission peaks by summing many coherent round trips, allowing high resolving power with a compact optical path. It can transmit a substantial fraction of resonant light when coating absorption is small, and works well for precise relative line separations and narrow-line measurements. A diffraction grating may require many illuminated grooves to achieve the same resolving power.

The etalon's free spectral range is limited, and repeated orders cause ambiguity across a broad optical spectrum. It commonly needs scanning and an order-selecting optical filter or another dispersive element. Mirror alignment, flatness and illumination angle strongly affect peak positions and widths, and absorption lowers throughput. A diffraction grating instead separates a broad range of wavelengths spatially for simultaneous detection, although grating orders can also overlap. The preferred instrument depends on whether narrow-band precision or broad spectral coverage is the priority.

A ring laser is a laser whose optical cavity follows a closed circulating path through a laser gain medium. Mirrors can fold the path into a triangle or another polygon; guided structures can implement the same circulating feedback principle. A travelling optical wave can return to its starting position and direction after one full circuit. The laser threshold requires round-trip gain to balance the complete losses; a closed path alone is not enough to produce laser oscillation.^[ref]

The phase condition explains the allowed normal modes. Neglect dispersion and take a constant round-trip reflection phase ${\varphi }_r$. If the circuit's optical path length is $L_{\mathrm{opt}}$, resonance requiresSubtracting the equations for consecutive modes gives the free spectral rangeMore generally the spacing is approximately the inverse round-trip group delay. A vacuum circuit of full geometric perimeter $P=0.90\mathrm{m}$ has a round-trip time about $3.00\mathrm{ns}$ and mode spacing about $333\mathrm{MHz}$. There is no additional factor of two: $P$ already counts the whole returning route.^[ref]

Gain and phase impose different conditions. In an original startup model let the three mirror intensity reflectivities be $0.99$, $0.99$ and $0.90$, and let the intensity gain accumulated during one complete circuit be $G=1.20$, with other losses neglected. The round-trip intensity multiplier isThe modeled field can grow from a seed. Reducing $G$ to $1.05$ gives $\mathcal{M}=0.9261945<1$, so it decays. These are intensity quantities, not amplitude reflectivities. Saturation and competition between normal modes must determine real steady operation.

A reciprocal ideal ring supports possible clockwise and counterclockwise waves. Their simultaneous stable emission is a dynamical question because both can interact with the same laser gain medium. A ring laser gyroscope uses rotation-dependent differences between the two circulation directions through the Sagnac effect; ring geometry by itself does not guarantee a useful rotation measurement.

A different motivating experiment used the directional dynamics as a nonlinear sensor. In 1988 Bruce McNamara, Kurt Wiesenfeld and Rajarshi Roy periodically modulated the asymmetry between counter-rotating lasing modes and added controlled external noise. Their measured signal-to-noise ratio improved for injected noise relative to no external noise, demonstrating stochastic resonance. This observation concerns a particular modulated dynamical response, not a claim that adding noise improves every optical measurement.^[ref]

A ring laser gyroscope measures a component of angular velocity from the frequency difference between two counter-circulating waves of a ring laser. Rotation changes their round-trip propagation times through the Sagnac effect. The active optical cavity converts this propagation difference into different resonant frequencies; combining the output waves gives an electrical beat signal through optical heterodyne detection.

For a planar evacuated ideal ring, let $P$ be its full perimeter, $\mathbf{A}$ its oriented vector area, $\mathbf{\Omega }$ its angular velocity and $\lambda$ the nominal vacuum wavelength. At small rotation speed the round-trip time difference has magnitude $|\Delta T|\simeq 4|\mathbf{A}\cdot \mathbf{\Omega }|/c^2$. With $T_0=P/c$ and nominal optical frequency ${\nu }_0=c/\lambda$, the resonant-frequency difference has magnitude ${\nu }_0|\Delta T|/T_0$. Choosing the output sign consistently with the area orientation givesThis is an ideal first-order scale law, with reciprocal paths and resolved counter-circulating modes. Optical dispersion and actual apparatus behaviour require a fuller response model. The underlying circulation-time difference is the one measured by a Sagnac interferometer.^[ref]

For an original square ring with $0.10\mathrm{m}$ sides, $A=0.010{\mathrm{m}}^2$ and $P=0.40\mathrm{m}$. At $\lambda =632.8\mathrm{nm}$ and normal-axis rotation $\Omega =0.010rad/s$,Tilting the ring changes the measured projection, not the full rotation magnitude. Reconstructing a general rotation vector requires independent directional information. The kilohertz-scale difference is also distinct from the optical carrier frequency and the cavity's free spectral range.

A historical experiment made the frequency comparison concrete. W. M. Macek and D. T. M. Davis, Jr. reported rotation sensing in 1963 using a continuous-wave helium-neon ring laser on the $1.153\mu \mathrm{m}$ line. Their metre-square apparatus used four gas tubes with Brewster windows and four external corner mirrors. The first-page diagram directs the two beams toward combining optics and a photocathode, linking the geometric circulation difference to a detectable beat.^[ref]

For two coherent output powers $I_1,I_2$, the ideal combined signal includesOne such intensity record supplies a beat-frequency magnitude, not automatically the rotation sign, since changing the sign can be absorbed into the unknown phase. A directional readout needs additional phase information or a suitable reference. Mode coupling, frequency bias and detector response must also be tested rather than treating a closed optical path as a complete calibrated sensor.