[4] Etalons are widely used in telecommunications, lasers and spectroscopy to control and measure the wavelengths of light.
In a typical system, illumination is provided by a diffuse source set at the focal plane of a collimating lens.
When properly designed, they run cooler than absorptive filters because they reflect unwanted wavelengths rather than absorbing them.
Semiconductor diode lasers sometimes use a true Fabry–Pérot geometry, due to the difficulty of coating the end facets of the chip.
Quantum cascade lasers often employ Fabry–Pérot cavities to sustain lasing without the need for any facet coatings, due to the high gain of the active region.
Stable Fabry–Pérot interferometers are often used to stabilize the frequency of light emitted by a laser (which often fluctuate due to mechanical vibrations or temperature changes) by means of locking it to a mode of the cavity.
Fabry–Pérot etalons can be used to prolong the interaction length in laser absorption spectrometry, particularly cavity ring-down, techniques.
An etalon of increasing thickness can be used as a linear variable optical filter to achieve spectroscopy.
[8] In gravitational wave detection, a Fabry–Pérot cavity is used to store photons for almost a millisecond while they bounce up and down between the mirrors.
This increases the time a gravitational wave can interact with the light, which results in a better sensitivity at low frequencies.
This principle is used by detectors such as LIGO and Virgo, which consist of a Michelson interferometer with a Fabry–Pérot cavity with a length of several kilometers in both arms.
Smaller cavities, usually called mode cleaners, are used for spatial filtering and frequency stabilization of the main laser.
If the two beams are out of phase, only a small portion of the launched light is stored inside the resonator.
are If there are no other resonator losses, the decay of light intensity per round trip is quantified by the outcoupling decay-rate constant
accumulates to[10] Resonances occur at frequencies at which light exhibits constructive interference after one round trip.
of modal index and wavenumber, respectively, physically representing opposite propagation directions, occur at the same absolute value
of the Lorentzian spectral line shape, we obtain expressed in terms of either the half-width-at-half-maximum (HWHM) linewidth
The response of the Fabry–Pérot resonator to an electric field incident upon mirror 1 is described by several Airy distributions (named after the mathematician and astronomer George Biddell Airy) that quantify the light intensity in forward or backward propagation direction at different positions inside or outside the resonator with respect to either the launched or incident light intensity.
can be obtained via the round-trip-decay approach[13] by tracing the infinite number of round trips that the incident electric field
within the free spectral range of the Fabry–Pérot resonator, whose adjacent peaks can be unambiguously distinguished spectroscopically, i.e., they do not overlap at their FWHM (see figure "The physical meaning of the Airy finesse").
The more general case of a Fabry–Pérot resonator with frequency-dependent mirror reflectivities can be treated with the same equations as above, except that the photon decay time
Also in this case each Airy distribution is the sum of all underlying mode profiles which can be strongly distorted.
Constructive interference occurs if the transmitted beams are in phase, and this corresponds to a high-transmission peak of the etalon.
If the transmitted beams are out-of-phase, destructive interference occurs and this corresponds to a transmission minimum.
Whether the multiply reflected beams are in phase or not depends on the wavelength (λ) of the light (in vacuum), the angle the light travels through the etalon (θ), the thickness of the etalon (ℓ) and the refractive index of the material between the reflecting surfaces (n).
The phase difference between each successive transmitted pair (i.e. T2 and T1 in the diagram) is given by[15] If both surfaces have a reflectance R, the transmittance function of the etalon is given by where is the coefficient of finesse.
The maximum reflectivity is given by and this occurs when the path-length difference is equal to half an odd multiple of the wavelength.
Thus where ℓ0 is The phase difference between the two beams is The relationship between θ and θ0 is given by Snell's law: so that the phase difference may be written as To within a constant multiplicative phase factor, the amplitude of the mth transmitted beam can be written as The total transmitted amplitude is the sum of all individual beams' amplitudes: The series is a geometric series, whose sum can be expressed analytically.
Due to the angle dependence of the transmission, the peaks can also be shifted by rotating the etalon with respect to the beam.
Another expression for the transmission function was already derived in the description in frequency space as the infinite sum of all longitudinal mode profiles.
Fabry–Pérot interferometer, using a pair of partially reflective, slightly wedged optical flats. The wedge angle is highly exaggerated in this illustration; only a fraction of a degree is actually necessary to avoid ghost fringes. Low-finesse versus high-finesse images correspond to mirror reflectivities of 4% (bare glass) and 95%.
A commercial Fabry–Pérot device
Electric fields in a Fabry–Pérot resonator.
[
10
]
The electric-field mirror reflectivities are
and
. Indicated are the characteristic electric fields produced by an electric field
incident upon mirror 1:
initially reflected at mirror 1,
launched through mirror 1,
and
circulating inside the resonator in forward and backward propagation direction, respectively,
propagating inside the resonator after one round trip,
transmitted through mirror 2,
transmitted through mirror 1, and the total field
propagating backward. Interference occurs at the left- and right-hand sides of mirror 1 between
and
, resulting in
, and between
and
, resulting in
, respectively.
Resonance enhancement in a Fabry–Pérot resonator.
[
10
]
(top) Spectrally dependent internal resonance enhancement, equaling the generic Airy distribution
. Light launched into the resonator is resonantly enhanced by this factor. For the curve with
, the peak value is at
, outside the scale of the ordinate. (bottom) Spectrally dependent external resonance enhancement, equaling the Airy distribution
. Light incident upon the resonator is resonantly enhanced by this factor.
Airy distribution
(solid lines), corresponding to light transmitted through a Fabry–Pérot resonator, calculated for different values of the reflectivities
, and comparison with a single Lorentzian line (dashed lines) calculated for the same
.
[
10
]
At half maximum (black line), with decreasing reflectivities the FWHM linewidth
of the Airy distribution broadens compared to the FWHM linewidth
of its corresponding Lorentzian line:
results in
, respectively.
Lorentzian linewidth and finesse versus Airy linewidth and finesse of a Fabry–Pérot resonator.
[
10
]
[Left] Relative Lorentzian linewidth
(blue curve), relative Airy linewidth
(green curve), and its approximation (red curve). [Right] Lorentzian finesse
(blue curve), Airy finesse
(green curve), and its approximation (red curve) as a function of reflectivity value
. The exact solutions of the Airy linewidth and finesse (green lines) correctly break down at
, equivalent to
, whereas their approximations (red lines) incorrectly do not break down. Insets: Region
.
The physical meaning of the Lorentzian finesse
of a Fabry–Pérot resonator.
[
10
]
Displayed is the situation for
, at which
and
, i.e., two adjacent Lorentzian lines (dashed colored lines, only 5 lines are shown for clarity for each resonance frequency,
) cross at half maximum (solid black line) and the Taylor criterion for spectrally resolving two peaks in the resulting Airy distribution (solid purple line, the sum of 5 lines which has been normalized to the peak intensity of itself) is reached.
The physical meaning of the Airy finesse
of a Fabry–Pérot resonator.
[
10
]
When scanning the Fabry–Pérot length (or the angle of incident light), Airy distributions (colored solid lines) are created by signals at individual frequencies. The experimental result of the measurement is the sum of the individual Airy distributions (black dashed line). If the signals occur at frequencies
, where
is an integer starting at
, the Airy distributions at adjacent frequencies are separated from each other by the linewidth
, thereby fulfilling the Taylor criterion for the spectroscopic resolution of two adjacent peaks. The maximum number of signals that can be resolved is
. Since in this specific example the reflectivities
have been chosen such that
is an integer, the signal for
at the frequency
coincides with the signal for
at
. In this example, a maximum of
peaks can be resolved when applying the Taylor criterion.
Example of a Fabry–Pérot resonator with (top) frequency-dependent mirror reflectivity and (bottom) the resulting distorted mode profiles
of the modes with indices
, the sum of 6 million mode profiles (pink dots, displayed for a few frequencies only), and the Airy distribution
.
[
10
]
The vertical dashed lines denote the maximum of the reflectivity curve (black) and the resonance frequencies of the individual modes (colored).
A Fabry–Pérot etalon. Light enters the etalon and undergoes multiple internal reflections.
The transmission of an etalon as a function of wavelength. A high-finesse etalon (red line) shows sharper peaks and lower transmission minima than a low-finesse etalon (blue).
Finesse as a function of reflectivity. Very high finesse factors require highly reflective mirrors.
Transient analysis of a silicon (
n
= 3.4) Fabry–Pérot etalon at normal incidence. The upper animation is for etalon thickness chosen to give maximum transmission while the lower animation is for thickness chosen to give minimum transmission.
False color transient for a high refractive index, dielectric slab in air. The thickness/frequencies have been selected such that red (top) and blue (bottom) experience maximum transmission, whereas the green (middle) experiences minimum transmission.
