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In optics, a Fabry–Pérot interferometer or etalon is typically made of a transparent plate with two reflecting surfaces, or two parallel highly reflecting mirrors. (Precisely, the former is an etalon and the latter is an interferometer, but the terminology is often used inconsistently.) Its transmission spectrum as a function of wavelength exhibits peaks of large transmission corresponding to resonances of the etalon. It is named after Charles Fabry and Alfred Perot, who developed the instrument in 1899. Etalon is from the French étalon, meaning "measuring gauge" or "standard".
Contents
- Basic description
- Applications
- 1 Resonator losses outcoupled light resonance frequencies and spectral line shapes
- 2 Generic Airy distribution The internal resonance enhancement factor
- 3 Other Airy distributions
- 4 Airy distribution as a sum of mode profiles
- 5 Characterizing the Fabry Prot resonator Lorentzian linewidth and finesse
- 6 Scanning the Fabry Prot resonator Airy linewidth and finesse
- 7 Frequency dependent mirror reflectivities
- 8 Description of the Fabry Perot resonator in wavelength space
- Detailed analysis
- Another expression for the transmission function
- References
Etalons are widely used in telecommunications, lasers and spectroscopy to control and measure the wavelengths of light. Recent advances in fabrication technique allow the creation of very precise tunable Fabry–Pérot interferometers.
Basic description
The heart of the Fabry–Pérot interferometer is a pair of partially reflective glass optical flats spaced micrometers to centimeters apart, with the reflective surfaces facing each other. (Alternatively, a Fabry–Pérot etalon uses a single plate with two parallel reflecting surfaces.) The flats in an interferometer are often made in a wedge shape to prevent the rear surfaces from producing interference fringes; the rear surfaces often also have an anti-reflective coating.
In a typical system, illumination is provided by a diffuse source set at the focal plane of a collimating lens. A focusing lens after the pair of flats would produce an inverted image of the source if the flats were not present; all light emitted from a point on the source is focused to a single point in the system's image plane. In the accompanying illustration, only one ray emitted from point A on the source is traced. As the ray passes through the paired flats, it is multiply reflected to produce multiple transmitted rays which are collected by the focusing lens and brought to point A' on the screen. The complete interference pattern takes the appearance of a set of concentric rings. The sharpness of the rings depends on the reflectivity of the flats. If the reflectivity is high, resulting in a high Q factor, monochromatic light produces a set of narrow bright rings against a dark background. A Fabry–Pérot interferometer with high Q is said to have high finesse.
Applications
1. Resonator losses, outcoupled light, resonance frequencies, and spectral line shapes
The spectral response of a Fabry-Pérot resonator is based on interference between the light launched into it and the light circulating in the resonator. Constructive interference occurs if the two beams are in phase, leading to resonant enhancement of light inside the resonator. If the two beams are out of phase, only a small portion of the launched light is stored inside the resonator. The transmitted and reflected light is spectrally modified compared to the incident light.
Assume a two-mirror Fabry-Pérot resonator of geometrical length
The electric-field and intensity reflectivities
If there are no other resonator losses, the photon-decay time
With
Resonances occur at frequencies at which light exhibits constructive interference after one round trip. Each resonator mode with its mode index
Two modes with opposite values
The decaying electric field at frequency
Fourier transformation of the electric field in time provides the electric field per unit frequency interval,
Each mode has a normalized spectral line shape per unit frequency interval given by
whose frequency integral is unity. Introducing the full-width-at-half-maximum (FWHM) linewidth
Calibrated to a peak height of unity, we obtain the Lorentzian lines:
When repeating the above Fourier transformation for all the modes with mode index
Since the linewidth
2. Generic Airy distribution: The internal resonance enhancement factor
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. The response of the Fabry-Pérot resonator is most easily derived by use of the circulating-field approach. This approach assumes a steady state and relates the various electric fields to each other (see figure "Electric fields in a Fabry-Pérot resonator").
The field
The generic Airy distribution, which considers solely the physical processes exhibited by light inside the resonator, then derives as the intensity circulating in the resonator relative to the intensity launched,
3. Other Airy distributions
Once the internal resonance enhancement, the generic Airy distribution, is established, all other Airy distributions can be deduced by simple scaling factors. Since the intensity launched into the resonator equals the transmitted fraction of the intensity incident upon mirror 1,
and the intensities transmitted through mirror 2, reflected at mirror 2, and transmitted through mirror 1 are the transmitted and reflected/transmitted fractions of the intensity circulating inside the resonator,
respectively, the other Airy distributions
The index “emit” denotes Airy distributions that consider the sum of intensities emitted on both sides of the resonator.
The back-transmitted intensity
It can be easily shown that in a Fabry-Pérot resonator, despite the occurrence of constructive and destructive interference, energy is conserved at all frequencies:
The external resonance enhancement factor (see figure "Resonance enhancement in a Fabry-Pérot resonator") is
At the resonance frequencies
Usually light is transmitted through a Fabry-Pérot resonator. Therefore, an often applied Airy distribution is
It describes the fraction
For
resulting in
Alternatively,
respectively. Exploiting
results in the same
4. Airy distribution as a sum of mode profiles
Physically, the Airy distribution is the sum of mode profiles of the longitudinal resonator modes. Starting from the electric field
and then sums over the emitted mode profiles of all longitudinal modes
thus equaling the Airy distribution
The same simple scaling factors that provide the relations between the individual Airy distributions also provide the relations among
5. Characterizing the Fabry-Pérot resonator: Lorentzian linewidth and finesse
The Taylor criterion of spectral resolution proposes that two spectral lines can be resolved if the individual lines cross at half intensity. When launching light into the Fabry-Pérot resonator, by measuring the Airy distribution, one can derive the total loss of the Fabry-Pérot resonator via recalculating the Lorentzian linewidth
The underlying Lorentzian lines can be resolved as long as the Taylor criterion is obeyed (see figure "The physical meaning of the Lorentzian finesse"). Consequently, one can define the Lorentzian finesse of a Fabry-Pérot resonator:
It is displayed as the blue line in the figure "The physical meaning of the Lorentzian finesse". The Lorentzian finesse
equivalent to
6. Scanning the Fabry-Pérot resonator: Airy linewidth and finesse
When the Fabry-Pérot resonator is used as a scanning interferometer, i.e., at varying resonator length (or angle of incidence), one can spectroscopically distinguish spectral lines at different frequencies within one free spectral range. Several Airy distributions
The Airy linewidth
The concept of defining the linewidth of the Airy peaks as FWHM breaks down at
For equal mirror reflectivities, this point is reached when
The finesse of the Airy distribution of a Fabry-Pérot resonator, which is displayed as the green curve in the figure "Lorentzian linewidth and finesse versus Airy linewidth and finesse of a Fabry-Pérot resonator" in direct comparison with the Lorentzian finesse
When scanning the length of the Fabry-Pérot resonator (or the angle of incident light), the Airy finesse quantifies the maximum number of Airy distributions created by light at individual frequencies
Often the unnecessary approximation
This approximation of the Airy linewidth, displayed as the red curve in the figure "Lorentzian linewidth and finesse versus Airy linewidth and finesse of a Fabry-Pérot resonator", deviates from the correct curve at low reflectivities and incorrectly does not break down when
7. Frequency-dependent mirror reflectivities
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
8. Description of the Fabry-Perot resonator in wavelength space
The varying transmission function of an etalon is caused by interference between the multiple reflections of light between the two reflecting surfaces. 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 - T1 in the diagram) is given by δ:
If both surfaces have a reflectance R, the transmittance function of the etalon is given by
where
is the coefficient of finesse.
Maximum transmission (
and this occurs when the path-length difference is equal to half an odd multiple of the wavelength.
The wavelength separation between adjacent transmission peaks is called the free spectral range (FSR) of the etalon, Δλ, and is given by:
where λ0 is the central wavelength of the nearest transmission peak and
This is commonly approximated (for R > 0.5) by
If the two mirrors are not equal, the finesse becomes
Etalons with high finesse show sharper transmission peaks with lower minimum transmission coefficients. In the oblique incidence case, the finesse will depend on the polarization state of the beam, since the value of "R", given by the Fresnel equations, is generally different for p and s polarizations.
A Fabry–Pérot interferometer differs from a Fabry–Pérot etalon in the fact that the distance ℓ between the plates can be tuned in order to change the wavelengths at which transmission peaks occur in the interferometer. Due to the angle dependence of the transmission, the peaks can also be shifted by rotating the etalon with respect to the beam.
Fabry–Pérot interferometers or etalons are used in optical modems, spectroscopy, lasers, and astronomy.
A related device is the Gires–Tournois etalon.
Detailed analysis
Two beams are shown in the diagram at the right, one of which (T0) is transmitted through the etalon, and the other of which (T1) is reflected twice before being transmitted. At each reflection, the amplitude is reduced by
where
The total amplitude of both beams will be the sum of the amplitudes of the two beams measured along a line perpendicular to the direction of the beam. The amplitude t0 at point b can therefore be added to t'1 retarded in phase by an amount
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:
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. The amplitude can be rewritten as
The intensity of the beam will be just t times its complex conjugate. Since the incident beam was assumed to have an intensity of one, this will also give the transmission function:
For an asymmetrical cavity, that is, one with two different mirrors, the general form of the transmission function is
Another expression for the transmission function
Defining
The second term is proportional to a wrapped Lorentzian distribution so that the transmission function may be written as a series of Lorentzian functions:
where