Fabry-Perot etalons are made-up of two partially reflecting mirrors parallel to each other. They are often used as frequency comb filters.

**1. INTRODUCTION**
Fabry-Perot etalons are narrow-band optical filters made-up of two partially reflecting mirrors parallel to each other at fixed distance. Depending of what is in between the mirrors, we can have glass etalons or air-gap etalons. Glass etalon is simply a plan-parallel glass plate of specific thickness and specific reflectivity mirror coatings. The air-gap etalon has a frame or spacers which hold mirrors at a fixed distance. In order to achieve high thermal and mechanical stability of air gap etalons, we use a glass with low thermal expansion coefficient, i.e., Zerodur® [1] as a spacer material. Air spaced etalons are easier to customize than solid glass etalons. When making air-gap etalons, mirrors from the same batch can be used for any gap. With glass etalons, each change in glass thickness requires a new coating runs.

Figure 1. Air spaced fiber pigtailed Fabry-Perot etalon manufactured by Doric Lenses Inc.

CAD model (left) and final product (right).

NOTE : This document is not intended to cover the theory on Fabry-Perot etalon, but as a reminder of some important properties. For a complete analysis on Fabry-Perot etalon we suggest the reference on resonator optics, Laser by A.E. Siegmann [2].

**2. TRANSMISSION OF FABRY-PEROT ETALON**
The transmitivity of a Fabry-Perot etalon is dependant of the wavelength of the incident radiation. The interference created between mirrors creates standing waves. Theses standing waves generate constructive interference when they fit exactly the cavity length. In other words, the cavity optical length has an integer multiple number of the resonnant wavelengths.

The transmitted intensity can then be expressed by the following equation :

(Eq. 1)

where : TMAX is the maximum transmission of resonance peaks

F is the finesse

n is the refractive index of the gap material,

d is the distance between mirrors,

λ is the wavelength

Figure 2. Spectral transmitivity of an ideal Fabry-Perot etalon

(R = 80%, FSR = 50.0 Ghz, ΔvFWHM = 3.57 Ghz, Finesse = 14).

In ideal etalons cavities, TMAX is equal to 1, but in real cavities with internal losses, surface roughness, mirrors tilt, etc it's different. TMAX can be expressed by :

(Eq. 2)

where: FT is the total finesse (defined in section 4)

FM is the mirror reflectivity finesse (defined in section 4)

**3. FREE SPECTRAL RANGE (FSR)**
Fabry-Perot etalon constitutes a resonator of a specific optical path length. The light is transmitted when there is constructive interference and is reflected back when there is destructive interference. There is constructive interference only if the optical path length is an integer number of the specific wavelength. So, for a specified optical path length, there is only discrete number of wavelength that fulfill this condition. That's why we observe a pattern of the transmission versus wave frequency (or wavelength) with resonnance peak transmission at specific frequency (see figure 2). [2,3]

The resonnance peaks are evenly spaced in spectral domain. Free spectral range (FSR) is defined as the spacing between two resonnance peaks in frequency domain. It can be shown that FSR is related to the cavity optical path length by the following equation :

(Eq. 3)

where: OPL=n×d is the Optical Path Length,

n is the refractive index of the gap material,

d is the distance between mirrors,

c0 is ligth speed in vacuum (299 792 458 m/sec).

Table 1. OPL associated to some standard FSR value.

**4. FINESSE**
Typical spectral transmission of a Fabry-Perot filter is shown in figure 2. We see equally spaced resonnance peaks. The finesse is related to the ratio of resonnance peak spectral separation over peaks spectral width. As you will find in reference [2] and [3], finesse is related to cavity losses. When loss are mainly due to mirrors reflectivity, a simple relation gives the value for finesse as follow :

(Eq. 4)

where: FM is finesse coefficient only due to mirrors reflectivity,

FWHM is peaks spectral width,

R1 and R2 are mirrors reflectivity.

For real etalons, the surface accuracy and paralelism affect finesse. Finesse coefficient can be expressed as :

where: FT is total finesse

FC is theoretical cavity finesse coefficient due to mirrors reflectivity and cavity losses,

FP is surface paralellism finesse coefficient,

FS is plate spherical deviation finesse coefficient,

Fθ is incident beam divergence dependant finesse coefficient,

FD is diffraction limited finesse coefficient,

R1 and R2 are mirrors reflectivity,

λ is central wavelength,

α is linear cavity loss coefficient (αd) is roundtrip loss,

ß is tilt angle between surfaces (typically 1 arcsec),

CA is etalon clear aperture diameter,

d is the distance between mirrors,

M is surface flat accuracy in fraction of wavelength (ex: for λ/100; M = 100),

θ is incident beam divergence.

**5. THERMAL STABILITY**
The temperature variation shifts the resonnance peak frequency in two ways. First, there is the thermal expansion of materials, specified by coefficient α. Second, there is the change of refractive index of glasses with temperature, or temperature dispersion, specified by coefficient; ∂n / ∂T.

Depending on type of glass used, one or both of these effects can lead to a problematic frequency shift. Several manufacturers produce glasses with low expansion coefficient and low thermal refractive index shift. To the best of our knowledge, no glass etalon achieves the stability of an air spaced etalon with Zerodur® holder. There are two ways to reduce frequency shift created by temperature variation. The fit way is to control the temperature of the etalon within a temperature controlled enclosure.

This is not always possible or practical solution. Second way is to have a fine tuning élément to compensate temperature variations. Fine tuning can be assured by a piezoelectric transducer with active feedback control loop. By controlling piezoelectric thickness, we control spacing between mirrors. Small change in mirrors spacing allows to scan the wavelength within a small range or to preserve a locked wavelength in spite of Small temperature variations.

The frequency shift per °C is given by the following relation :

(Eq. 11)

where : c0 is light speed (299 792 458 m/sec),

λ0 is resonnance wavelength,

n is refractive index of the gap material,

∂n / ∂T is the temperature dispersion of gap material,

α is thermal expansion coefficient of gap material.

Table 2. Temperature variations of Fabry-Perot etalons

**6. REFERENCES**
1. Zerodur® glass is a registred trademark of SCHOTT AG, Mainz, Germany.

2. A.E. Siegmann, '' Lasers '', University Sciences Books, Mills Valley, 1986, chapter 11.

3. B.E.A. Saleh & M.C. Teich, '' Fundamental of Photonics '', Wiley-Interscience, New-York,

1991, chapter 9.

Written by Sead Doric and Jean-Luc Néron from Doric Lenses Inc.
Updated March 23th 2005.