# Etalon

An etalon is a optical devise inserted into a laser cavity in order to extend the coherence length. It works by selecting a single frequency of the cavity and only allowing it to propegate. The other modes present are extiguished.

Etalon redirects here. Etalon is also the French word for stallion.

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 Pérot. 'Etalon' is from the French étalon, meaning 'measuring gauge' or 'standard' [1].

Etalons are widely used in telecommunications, lasers and spectroscopy for controlling and measuring the wavelength of light. Recent advances in fabrication technique allow the creation of very precise tunable Fabry-Pérot interferometers. Fabry-Pérot interferometers also form the most common type of optical cavity used in laser construction.

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.

## Theory

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, 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:

${\displaystyle \delta =\left({\frac {2\pi }{\lambda }}\right)2nl\cos \theta .}$

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).

If both surfaces have a reflection coefficient R, the transmission function of the etalon is given by:

${\displaystyle T_{e}={\frac {(1-R)^{2}}{1+R^{2}-2R\cos(\delta )}}}$

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 reflectivity of the etalon Re is the complement of the transmission, such that Te + Re = 1. The maximum reflectivity is given by:

${\displaystyle R_{max}={\frac {4R}{(1+R)^{2}}}}$

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.

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

${\displaystyle \Delta \lambda ={\frac {\lambda _{0}^{2}}{2nl\cos \theta }}}$

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:

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

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

This is commonly approximated (for R > 0.5) by

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

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. Due to the angle dependence of the transmission, the peaks can also be shifted by rotating the etalon with respect to the beam.

### Detailed analysis

Two beams are shown in the diagram at the right, one of which (${\displaystyle T_{0}}$ ) is transmitted through the etalon, and the other of which (${\displaystyle T_{1}}$ ) is reflected twice before being transmitted. At each reflection, the amplitude is reduced by ${\displaystyle {\sqrt {R}}}$  and the phase is shifted by ${\displaystyle \pi }$ , while at each transmission through an interface the amplitude is reduced by ${\displaystyle {\sqrt {T}}}$ . Assuming no absorption, we have by conservation of energy ${\displaystyle T+R=1}$ . Define n as the index of refraction inside the etalon, and ${\displaystyle n_{0}}$  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 ${\displaystyle a}$  is unity. The amplitude at point ${\displaystyle b}$  will then be

${\displaystyle T_{0}=T\,e^{ikl/\cos \theta }}$

where ${\displaystyle k=2\pi /\lambda }$  is the wave number inside the etalon. At point ${\displaystyle c}$  the amplitude will be

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

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 ${\displaystyle T_{1}}$  equal in magnitude to the amplitude at point c, but which has been retarded in phase by an amount ${\displaystyle k_{0}l_{0}}$  where ${\displaystyle k_{0}=2\pi n_{0}/\lambda }$  is the wave number outside of the etalon. Thus:

${\displaystyle T_{1}=RT\,e^{2\pi i+3ikl/\cos \theta -ik_{0}l_{0}}}$

where ${\displaystyle l_{0}}$  is seen to be:

${\displaystyle l_{0}=2l\tan(\theta )\sin(\theta _{0})\,}$

Neglecting the ${\displaystyle 2\pi }$  phase change due to the two reflections, we have for the phase difference between the two beams

${\displaystyle \delta =2kl/\cos(\theta )-2k_{0}l_{0}\,}$

The relationship between ${\displaystyle \theta }$  and ${\displaystyle \theta _{0}}$  is given by Snell's law:

${\displaystyle n\sin(\theta )=n_{0}\sin(\theta _{0})\,}$

So that the phase difference may be written

${\displaystyle \delta =2nkl\,\cos(\theta )\,}$

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

${\displaystyle T_{m}=TR^{m}e^{im\delta }\,}$

The total transmitted beam is the sum of all individual beams

${\displaystyle A_{T}=\sum _{m=0}^{\infty }T_{m}=T\sum _{m=0}^{\infty }R^{m}\,e^{im\delta }}$

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

${\displaystyle A_{T}={\frac {T}{1-Re^{i\delta }}}}$

The intensity of the beam will be just ${\displaystyle A_{T}A_{T}^{*}}$  and, since the incident beam was assumed to have an intensity of unity, this will also give the transmission function:

${\displaystyle T_{e}=A_{T}A_{T}^{*}={\frac {T^{2}}{1+R^{2}-2R\cos(\delta )}}}$

### 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.

Another useful expression for the transmission function may be derived as follows: The sum representation of the amplitude ${\displaystyle A_{T}}$   may be used directly to express the transmission function:

${\displaystyle T_{e}=T^{2}\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }R^{m+n}\,e^{i(m-n)\delta }}$

Defining ${\displaystyle l=m-n}$ , rearranging terms, and using the geometric series formula on R yields

${\displaystyle T_{e}={\frac {T^{2}}{1-R^{2}}}\sum _{l=-\infty }^{\infty }R^{|l|}\,e^{il\delta }}$

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

${\displaystyle T_{e}={\frac {T^{2}}{1-R^{2}}}\sum _{l=-\infty }^{\infty }\int _{-\infty }^{\infty }L(\delta -\delta ';\gamma )\,e^{il\delta '}\,d\delta '}$

where ${\displaystyle L(x,\gamma )}$  is the Lorentz distribution:

${\displaystyle L(x;\gamma )\equiv {\frac {\gamma }{\pi (x^{2}-\gamma ^{2})}}}$

and ${\displaystyle \gamma \equiv \ln(1/R)}$ . 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 ${\displaystyle D_{T}(x)}$  and so the transmission function is seen to be the convolution of a Lorentzian function and a Dirac comb:

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