Fabry-Pérot interferometer

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"Étalon" redirects here. For the commune in Somme, France, see Étalon, Somme.

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. (Technically 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.[1] "Etalon" is from the French étalon, meaning "measuring gauge" or "standard".[2]

The resonance effect of the Fabry-Pérot interferometer is identical to that used in a dichroic filter. That is, dichroic filters are very thin sequential arrays of Fabry-Pérot interferometers, and are therefore characterised and designed using the same mathematics.

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.

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[edit] Theory

A Fabry-Pérot etalon. Light enters the etalon and undergoes multiple internal reflections.
A Fabry-Pérot etalon. Light enters the etalon and undergoes multiple internal reflections.

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 succeeding reflection is given by δ:

\delta = \left( \frac{2 \pi}{\lambda} \right) 2 n l \cos\theta.

If both surfaces have a reflectance R, the transmittance function of the etalon is given by:

T_e = \frac{(1-R)^2}{1+R^2-2R\cos(\delta)}=\frac{1}{1+F\sin^2(\frac{\delta}{2})}

where F = \frac{4R}{{(1-R)^2}} is the coefficient of finesse.

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).
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).

Maximum transmission (Te = 1) occurs when the optical path length difference (2nl cos θ) between each transmitted beam is an integer multiple of the wavelength. In the absence of absorption, the reflectance of the etalon Re is the complement of the transmittance, such that Te + Re = 1. The maximum reflectivity is given by:

R_{max} = 1-\frac{1}{1+F}= \frac {4R}{(1+R)^2}

and this occurs when the path-length difference is equal to half an odd multiple of the wavelength.

Finesse as a function of reflectivity. Very high finesse factors require highly reflective mirrors.
Finesse as a function of reflectivity. Very high finesse factors require highly reflective mirrors.

The wavelength separation between adjacent transmission peaks is called the free spectral range (FSR) of the etalon, Δλ, and is given by:

\Delta\lambda = \frac{ \lambda_0^2}{2nl \cos\theta + \lambda_0 } \approx \frac{ \lambda_0^2}{2nl \cos\theta }

where λ0 is the central wavelength of the nearest transmission peak. The FSR is related to the full-width half-maximum, δλ, of any one transmission band by a quantity known as the finesse:

\mathcal{F} = \frac{\Delta\lambda}{\delta\lambda}=\frac{\pi}{2 \arcsin(1/\sqrt F)}.

This is commonly approximated (for R > 0.5) by

\mathcal{F} \approx \frac{\pi \sqrt{F}}{2}=\frac{\pi R^{1/2} }{(1-R)}

Etalons with high finesse show sharper transmission peaks with lower minimum transmission coefficients.

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.

[edit] 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 \sqrt{R} and the phase is shifted by π, while at each transmission through an interface the amplitude is reduced by \sqrt{T}. Assuming no absorption, we have by conservation of energy T + R = 1. Define n as the index of refraction inside the etalon, and n0 as the index of refraction outside the etalon. Using phasors to represent the amplitude of the radiation, let's suppose that the amplitude at point a is unity. The amplitude at point b will then be

T_0=T\,e^{ikl/\cos\theta}

where k = 2πn / λ is the wave number inside the etalon and λ is the vacuum wavelength. At point c the amplitude will be

TR\,e^{2\pi i + 3ikl/\cos\theta}

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. We therefore add the amplitude at point b to an amplitude T1 equal in magnitude to the amplitude at point c, but which has been retarded in phase by an amount k0l0 where k0 = 2πn0 / λ is the wave number outside of the etalon. Thus:

T_1=RT\,e^{2\pi i+3ikl/\cos\theta-ik_0l_0}

where l0 is seen to be:

l_0=2l\tan(\theta)\sin(\theta_0)\,

Neglecting the 2π phase change due to the two reflections, we have for the phase difference between the two beams

\delta={2kl\over\cos(\theta)}-k_0l_0\,

The relationship between θ and θ0 is given by Snell's law:

n\sin(\theta)=n_0\sin(\theta_0)\,

So that the phase difference may be written

\delta=2kl\,\cos(\theta)\,

To within a constant multiplicative phase factor, the amplitude of the m-th transmitted beam can be written as

T_m=TR^m e^{im\delta}\,

The total transmitted beam is the sum of all individual beams

A_T=\sum_{m=0}^\infty T_m=T\sum_{m=0}^\infty R^m\,e^{im\delta}

The series is a geometric series whose sum can be expressed analytically. The amplitude can be rewritten as

A_T=\frac{T}{1-Re^{i\delta}}

The intensity of the beam will be just A_TA_T^* and, since the incident beam was assumed to have an intensity of unity, this will also give the transmission function:

T_e=A_TA_T^*=\frac{T^2}{1+R^2-2R\cos(\delta)}

[edit] Another expression for the transmission function

A picture of the solar corona taken with the LASCO C1 coronagraph which employed a tunable Fabry-Pérot interferometer to recover scans of the solar corona at a number of wavelengths near the FeXIV green line at 5308 Å. The picture is a color coded image of the doppler shift of the line, which may be associated with the coronal plasma velocity towards or away from the satellite camera. In calculating the velocity, the velocity due to solar rotation has been subtracted.
A picture of the solar corona taken with the LASCO C1 coronagraph which employed a tunable Fabry-Pérot interferometer to recover scans of the solar corona at a number of wavelengths near the FeXIV green line at 5308 Å. The picture is a color coded image of the doppler shift of the line, which may be associated with the coronal plasma velocity towards or away from the satellite camera. In calculating the velocity, the velocity due to solar rotation has been subtracted.

Another useful expression for the transmission function may be derived as follows: The sum representation of the amplitude AT  may be used directly to express the transmission function:

T_e=T^2\sum_{m=0}^\infty \sum_{n=0}^\infty R^{m+n}\,e^{i(m-n)\delta}

Defining l = m − n, rearranging terms, and using the geometric series formula on R yields

T_e=\frac{T^2}{1-R^2}\sum_{l=-\infty}^\infty R^{|l|}\,e^{il\delta}

The terms of the sum are seen to be the characteristic function of the Lorentz distribution which allows the sum to be written:

T_e=\frac{T^2}{1-R^2}\sum_{l=-\infty}^\infty \int_{-\infty}^\infty L(\delta-\delta';\gamma)\,e^{il\delta'}\,d\delta'

where L(x,γ) is the Lorentz distribution:

L(x;\gamma) = \frac{\gamma}{\pi (x^2-\gamma^2)}

and γ = ln(1 / R). The order of integration and summation may be interchanged which yields a sum over the exponential term alone. This sum is seen to be a Dirac comb DT(x) and so the transmission function is seen to be the convolution of a Lorentzian function and a Dirac comb:

T_e=\frac{2\pi\,T^2}{1-R^2}\,\,L(\delta;\gamma)*D_{2\pi}(\delta)

[edit] Applications

  • The most important common applications are as dichroic filters, in which a series of etalonic layers are deposited on an optical surface by vapor deposition. These optical filters usually have more exact reflective and pass bands than absorptive filters. When properly designed, they run cooler than absorptive filters because they can reflect unwanted wavelengths. Dichroic filters are widely used in optical equipment such as light sources, cameras and astronomical equipment.
  • Telecommunications networks employing wavelength division multiplexing have add-drop multiplexers with banks of miniature tuned fused silica or diamond etalons. These are small iridescent cubes about 2 mm on a side, mounted in small high-precision racks. The materials are chosen to maintain stable mirror-to-mirror distances, and to keep stable frequencies even when the temperature varies. Diamond is preferred because it has greater heat conduction and still has a low coefficient of expansion. In 2005, some telecommunications equipment companies began using solid etalons that are themselves optical fibers. This eliminates most mounting, alignment and cooling difficulties.
  • A wavemeter is a combination of up to five Fabry-Pérot interferometers with a factor of ten difference in Δλ between any two of them. The beam is made divergent by a cylindrical lens and the distance between two bright lines is recorded by means of a CCD camera.
  • Laser resonators are often described as Fabry-Pérot resonators, although for many types of laser the reflectivity of one mirror is close to 100%, making it more similar to a Gires-Tournois interferometer. Semiconductor diode lasers sometimes use a true Fabry-Pérot geometry, due to the difficulty of coating the end facets of the chip.
  • Etalons are used to construct single-mode lasers. Without an etalon, a laser will generally produce light over a wavelength range corresponding to a number of cavity modes, which are similar to Fabry-Pérot modes. Inserting an etalon into the laser cavity, with well-chosen finesse and free-spectral range, can suppress all cavity modes except for one, thus changing the operation of the laser from multi-mode to single-mode.
  • In astronomy an etalon is used to select a single frequency of an atomic transition for imaging. The most common is the H-alpha line of the sun. Etalons are available to visual astronomers with bandwidths (FSR) from 0.2 nm to 1 nm. The CaK line from the sun is also commonly imaged with etalons in the 1 nm to 2 nm range.

[edit] See also

[edit] Notes and references

  1. ^ Perot frequently spelled his name with an accent—Pérot—in scientific publications, and so the name of the interferometer is commonly written with the accent.
    • Métivier, Françoise (September–October 2006). "Jean-Baptiste Alfred Perot" (in French) (pdf). Photoniques (no. 25).  Page 2: "Pérot ou Perot?"
  2. ^ Oxford English Dictionary