Etalon

An etalon is a Fabry-Perot interferometer formed by two parallel partially-reflective surfaces (often integrated as a single solid glass plate with reflective coatings on each face). Light enters one face, multiply-reflects between the two surfaces, and exits the other face. Only wavelengths satisfying the round-trip phase condition are transmitted efficiently; other wavelengths are largely reflected.

Operating principle. The transmission $T(\nu )$ as a function of frequency is the Airy distribution:

$$T(\nu )=\frac{(1-R{)}^2}{(1-R{)}^2+4R{\sin }^2(\varphi \mathrm{/}2)}$$ $$\varphi =\frac{4\pi nd\cos \theta }{\lambda }$$

where $R$ is the mirror power reflectivity, $n$ is the etalon refractive index, $d$ is the spacing, $\theta$ is the angle inside the etalon, and $\lambda$ is the vacuum wavelength.

Transmission is maximum (= 1) when $\varphi$ is an integer multiple of $2\pi$, and minimum when $\varphi$ is an odd multiple of $\pi$.

Free spectral range and finesse. The frequency spacing between adjacent transmission peaks is the free spectral range:

$$\mathrm{\Delta }{\nu }_{\text{FSR}}=\frac{c}{2n_{g}d\cos \theta },$$

where $n_g$ is the group index of the spacer rather than its refractive index, because the index differs slightly from one peak to the next: 1.463 for fused silica at 1550 nm against a refractive index of 1.444, and 1 for air.

For mirrors of high reflectance the width of each transmission peak (FWHM) is

$$\mathrm{\Delta }\nu =\frac{\mathrm{\Delta }{\nu }_{\text{FSR}}}{F},F=\frac{\pi \sqrt{R}}{1-R},$$

where $F$ is the finesse. High-finesse etalons have narrow transmission peaks and broad transmission gaps. The exact width of the Airy peak is $(2\mathrm{\Delta }{\nu }_{\text{FSR}}\mathrm{/}\pi )\arcsin [(1-R)\mathrm{/}(2\sqrt{R})]$, 2% wider than $\mathrm{\Delta }{\nu }_{\text{FSR}}\mathrm{/}F$ at $R$ = 0.5 and 8.5% wider at 0.3; below $R=3-2\sqrt{2}\approx 0.17$ the transmission never falls to half its peak, and the width is undefined.

Standard etalon types.

Applications.

1. Narrowband optical filters: pass only specific wavelengths; reject all others. Used in:

   • Wavelength-division multiplexing receivers
   • Astronomical instruments (selecting specific atomic lines)
   • Atmospheric and Raman spectroscopy
   • Solar telescopes (selecting H-alpha line)

2. Wavelength references: etalon transmission peaks at ITU grid wavelengths serve as fixed-frequency markers for locking tunable lasers. A 50 GHz FSR etalon provides a ITU 50 GHz grid reference.

3. Intracavity laser tuning: inserting an etalon into a laser cavity restricts oscillation to specific frequencies, enabling single-longitudinal-mode operation in lasers that otherwise lase multimode.

4. Frequency discriminators: the steep transmission edge of an etalon converts frequency changes into intensity changes, enabling fast frequency-error signal generation for laser stabilization.

5. Spectrum analyzers: a scanning etalon (Fabry-Perot interferometer) can resolve laser linewidth and longitudinal-mode structure with very high resolution (well below the resolution of a typical OSA).

Solid glass etalons in detail. The most common laboratory etalon: a polished glass plate with $\sim 30\mathrm{\%}$ reflectivity coatings on both faces. Typical specifications:

Air-spaced etalons. Higher-stability applications use Invar or ULE (ultra-low expansion) spacers separating two precision-polished mirrors. These offer:

• Adjustable spacing (via heat or piezo) for tunable filter applications
• Higher temperature stability
• Higher achievable finesse: up to about ${10}^4$ for tunable etalons, and a few times ${10}^5$ for ULE reference cavities
• Lower bulk material absorption

Standard examples: laser frequency-stabilization cavities for atomic-clock applications, ULE-spaced reference cavities used by the SI second's optical-clock implementations.

On-chip etalons (ring resonators). Silicon photonic microring resonators are 2D-confined etalons, with a finesse of typically 50 – 1,000 in silicon and up to about 30,000 in low-loss silicon nitride. Used for:

• Add-drop wavelength filters
• WDM channel multiplexing
• Microring modulators
• Optical sensors (refractive index, biosensing)
• Frequency comb generators

Etalon distortion (parallelism). Imperfect surface parallelism degrades finesse: the device acts as a continuum of slightly different etalons across its aperture. Standard fabrication tolerance is $<1$ arcsec parallelism for high-finesse devices.

Etalon ghosts in laser systems. Stray reflections between parallel surfaces in a laser path can form an unintended etalon, creating fringes in the laser intensity vs wavelength. Common sources: input/output windows of cuvettes, parallel-faced detectors, BS surfaces. The standard fix is to slightly wedge the surfaces (1 – 5° angle) to prevent forming a finite-aperture etalon.

Distinguishing etalon from Fabry-Perot resonator. "Etalon" typically refers to a passive filter or measurement device; "Fabry-Perot" can refer to either an active laser cavity or a passive etalon. The mathematics is identical; the names reflect historical and applied-physics convention rather than physical distinction.

The transmission comb of a solid etalon (a free-standing plate, or a plate between any two media) is one of the presets in the Thin-Film Reflectance Calculator, which draws the fringes from the same characteristic matrix method it uses for coatings and checks itself against the Airy formula above. The Fabry-Perot Cavity Calculator works the other way round, from the mirror reflectivities, length, index and loss to the free spectral range, finesse, linewidth, Q, photon lifetime and fringe contrast, with coated fused-silica and air-spaced etalons among its presets.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 11 (Fabry-Perot resonators and etalons); Hecht, Optics (5th ed., 2017), Ch. 9 for the standard treatment; Born & Wolf, Principles of Optics (7th ed., 1999), Ch. 7.