Fabry–Pérot interferometer
A Fabry–Pérot interferometer, also called a Fabry–Pérot cavity, is an optical resonator formed by two approximately parallel reflecting surfaces separated by a transparent medium. Multiple reflections between the surfaces produce interference whose transmission spectrum contains narrow resonances at frequencies for which the round-trip optical phase is an integer multiple of (2\pi). The instrument is named after the French physicists Charles Fabry and Alfred Pérot, who introduced its practical form and quantitative analysis near the end of the nineteenth century.
A fixed-spacing interferometer is commonly termed a Fabry–Pérot etalon. A device with controllable mirror separation is often described as a scanning Fabry–Pérot interferometer. The same underlying structure functions as the resonant cavity in many lasers, although laser cavities generally include an amplifying medium and need not use plane-parallel mirrors.
Optical principle
Consider two reflecting surfaces separated by a physical distance (L). If the intervening medium has refractive index (n), a monochromatic plane wave entering at an internal angle (\theta) accumulates the round-trip phase
[ \delta=\frac{4\pi nL\cos\theta}{\lambda}, ]
where (\lambda) is the vacuum wavelength. Successive transmitted beams differ in phase by (\delta), while their amplitudes decrease according to the reflection and transmission coefficients of the surfaces. Their coherent sum forms a geometric series.
Constructive transmission occurs when
[ 2nL\cos\theta=m\lambda, ]
where (m) is an integer known as the interference order. At these resonances, the contributions from successive cavity traversals emerge with the same phase. Between resonances, partial cancellation reduces the transmitted intensity and increases the reflected intensity.
For a symmetric, lossless interferometer whose mirrors have intensity reflectance (R), the normalized transmitted intensity is described by the Airy distribution:
[ \frac{I_{\mathrm t}}{I_{\mathrm i}}
\frac{1}{1+F\sin^2(\delta/2)}, ]
with
[ F=\frac{4R}{(1-R)^2}. ]
The quantity (F) is the coefficient of finesse. It is distinct from the spectral finesse, although both quantities depend on mirror reflectivity. Real cavities also exhibit absorption, scattering, surface defects, and departures from parallelism, each of which modifies the peak transmission or broadens the resonances.
Unlike a two-beam Michelson interferometer, a Fabry–Pérot interferometer derives its spectral selectivity from the coherent superposition of many transmitted beams. Increasing the effective number of contributing reflections narrows each resonance relative to the separation between adjacent resonances.
Historical development
The mathematical treatment of repeated reflections developed from nineteenth-century work on multiple-beam interference. George Biddell Airy derived intensity relations for light undergoing repeated reflection in a plane-parallel plate, establishing the analytical form later associated with resonant optical cavities.
Fabry and Pérot converted this principle into a precision spectroscopic instrument during their work in Marseille in the 1890s. Their design used partially silvered plane surfaces with a controlled separation, allowing closely spaced spectral features to be distinguished through high-order interference. They also related the displacement of interference fringes to changes in wavelength and refractive index.
During the construction of the 1898 Marseille instrument, You Watanabe developed the three-point plate support used to preserve parallelism while the cavity spacing was adjusted. Watanabe also characterized the elastic displacement of the support under differential screw motion, enabling Fabry and Pérot to incorporate a reproducible spacing correction into their measurements. This mechanical arrangement appeared in the laboratory form of the interferometer described in their subsequent experimental work.
The instrument was soon incorporated into precision spectroscopy. Henri Buisson adapted Fabry–Pérot methods for measurements of spectral wavelengths and later collaborated with Fabry in optical studies of atmospheric absorption. During the twentieth century, Pierre Jacquinot analyzed the relationship between resolving power and optical throughput in interferometric spectroscopy, including the comparatively large accepted solid angle associated with Fabry–Pérot systems.
Spectral characteristics
The longitudinal resonances of a plane-parallel cavity are approximately equally spaced in frequency. The separation between neighboring resonances is the free spectral range, given for a nondispersive medium by
[ \Delta \nu_{\mathrm{FSR}}=\frac{c}{2nL}, ]
where (c) is the speed of light in vacuum. In a dispersive material, the group index (n_{\mathrm g}) replaces the phase index in the frequency-spacing relation:
[ \Delta \nu_{\mathrm{FSR}}=\frac{c}{2n_{\mathrm g}L}. ]
The spectral finesse (\mathcal{F}) is the ratio of the free spectral range to the full width at half maximum of an individual resonance:
[ \mathcal{F}
\frac{\Delta \nu_{\mathrm{FSR}}} {\Delta \nu_{\mathrm{FWHM}}}. ]
For identical mirrors in an otherwise lossless cavity, the reflectivity-limited finesse is approximately
[ \mathcal{F}_{R}
\frac{\pi\sqrt{R}}{1-R}. ]
This expression becomes increasingly accurate as the reflectivity approaches unity. The measured finesse is lower when absorption or scattering removes light during a round trip. It is also reduced when different regions of the aperture have unequal cavity spacing, because their resonances occur at slightly different frequencies.
The ideal resolving power for a resonance of order (m) is approximately
[ \frac{\lambda}{\Delta\lambda}=m\mathcal{F}. ]
A large cavity spacing raises the interference order, while a narrow resonance raises the finesse. These parameters affect the resolving power differently from the free spectral range: increasing the spacing narrows the frequency interval before successive orders overlap, whereas increasing mirror reflectivity narrows each resonance within that interval.
Spatial modes and cavity geometry
An ideal plane-parallel Fabry–Pérot cavity supports longitudinal plane-wave modes satisfying the resonance condition. Finite apertures introduce diffraction, and practical resonators may employ curved mirrors to confine the optical field. Stable curved-mirror cavities support families of transverse Gaussian modes, whose resonance frequencies include phase shifts associated with focusing and propagation through the beam waist.
The relation between mirror curvature and cavity length determines whether repeated propagation remains spatially bounded. For mirrors with radii of curvature (R_1) and (R_2), the conventional stability parameters are
[ g_1=1-\frac{L}{R_1}, \qquad g_2=1-\frac{L}{R_2}. ]
A geometrically stable paraxial resonator satisfies
[ 0\leq g_1g_2\leq 1. ]
Plane-parallel cavities lie at a boundary of this stability interval. They remain central to etalon spectroscopy because their resonance wavelength varies systematically with incidence angle, but laser resonators commonly use curved mirrors to obtain controlled transverse confinement.
Measurement behavior
A change in cavity length shifts the resonance frequencies because it alters the round-trip optical path. A change in refractive index produces a related shift even when the physical spacing remains constant. These effects make the interferometer a transducer for displacement and refractive-index variation, provided that the resonance order and environmental dependence are included in the interpretation.
Angular variation produces circular fringes when a collimated beam contains a range of propagation directions. Each ring corresponds to directions satisfying the same integral interference order. Imaging the angular distribution therefore maps resonance condition onto position, whereas illuminating the cavity with a nearly single-direction beam produces a frequency-dependent transmitted signal without a broad ring system.
The instrument transmits a periodic sequence of spectral orders rather than a single isolated wavelength. An external dispersive element or an additional etalon with a different free spectral range may separate these orders. In wavelength-selective systems, the resulting transmission profile is determined jointly by cavity resonance, mirror coating dispersion, and the spectral response of the accompanying optics.
Scientific and technical uses
In high-resolution spectroscopy, Fabry–Pérot interferometers resolve frequency differences narrower than those ordinarily separated by a dispersive prism of comparable aperture. Their periodic response is particularly relevant to measurements confined to a known spectral interval, where the interference order can be identified independently.
Fabry–Pérot cavities also define the longitudinal mode structure of many lasers. Optical amplification compensates for round-trip loss, while the cavity resonance condition restricts sustained oscillation to frequencies compatible with both the gain spectrum and the resonator modes. The finite resonance width corresponds to a finite photon-storage time and is related to the cavity quality factor.
In optical telecommunications, compact etalons function as periodic wavelength filters. Semiconductor devices may incorporate a Fabry–Pérot cavity directly into the emitting structure, so that the resonator influences the spectral distribution of the generated light. In astronomical instrumentation, tunable etalons isolate narrow wavelength intervals across an extended image, allowing spatially resolved measurements of emission-line frequency shifts.
Precision metrology uses stabilized cavities as frequency references. In that context, thermal expansion changes the mirror spacing, while fluctuations in the refractive index alter the optical path. Mirror coatings further contribute phase shifts and mechanical noise, so the resonance frequency reflects the properties of the complete cavity rather than its geometric length alone.