The Fabry Perot interferometer consists of two parallel flat semi-transparent mirrors separated by a fixed distance. Light that enters the etalon undergoes multiple reflections and the interference of the light emerging from the etalon during each bounce causes a modulation in the transmitted and reflected beams. During one return bounce the phase changes by 2px2ndCOS(theta)/lambda, where lambda is the angle of the beam in the etalon. Constructive and destructive interference occurs based on the angle of the beam (lambda) the optical thickness of the etalon (nd) and the wavelength(lambda).
The transmission spectrum of an etalon will have a series of peaks, where constructive interference occurs, spaced by the 'free spectral range' or FSR. As seen on the right in our online etalon designer. If the absorption and scattering losses are small, the reflection spectrum of the etalon is 1 - T.
FSR = 1/2nd cm-1 (wave number)
FSR = lambda2 /2nd nm (wavelength)
FSR = c/2nd Hz (frequency)
You will notice that the mirror reflectivity is not part of these equations. The mirror reflectivity does not affect the FSR, it affects the number of bounces and improves the quality of the modulation (more perfect bounces = better modulation). etalon-calc-small.jpg As the reflectivity of the mirrors is increased the modulation peaks become sharper and decrease in width. The full width at half maximum of the peaks is called the bandwidth and the ratio of the line width to the distance between the peaks (the FSR) is called the Finesse, F. So the Bandwidth = FSR / F. And the finesse is the ratio of the FSR to the Bandwdth, F = FSR / Bandwidth.
Bandwidth is the full width at half maximum (FWHM) of the peak
Bandwidth = FSR / F
The finesse is a dimensionless quantity and the units of the Bandwidth are the same as the FSR.
Another quantity, The Coefficient of Finesse, F, = 4R/(1-R)^2 and the maximum reflectivity of the etalon is Rmax = 4R/(1+R)^2. The Coefficient of Finesse and the finesse are related by the equation F=PI/(2arcsin(1/SQRT(F))) which can be approximated to F=(PI/SQRT(F))/2 or
F=PI/SQRT(4R/(1-R)^2).
Actual versus Theorectical Performance
Etalons are usually described in terms of FSR and finesse. In many textbooks, the finesse is calculated using only the parameter R, reflectivity of the mirrors as in F=PI/SQRT(4R/(1-R)^2). The etalon is assumed to have no loses, like scatter or imperfect surface flatness. This is like our physics class questions that started by assuming "a mass is sliding down a frictionless incline...". So a perfect etalon with no losses or imperfections will always have a 100% peak transmission.
Limits to Finesse
In reality there are other factors that 'limit' the transmission and finesse such as surface irregularity, parallelism, coating scatter. Each one makes a contribution to limiting the finesse and then all these contributions are combined to come up with the expected finesse and transmission. In our etalon calculator the graph displays both the perfect theoretical transmission and the expected transmission taking into account all the defects in a real etalon.
Surface Figure
Surface figure is the rms variation of surface away from flat. Spherical error is excluded from the rms surface figure and is treated separately. Surface figure is usually measured at the HeNe laser wavelength of 633nm and is expressed as fractions of this wavelength. The numbers that are included in the calculator are examples of practical values such as;. 633/20 = 30nm and a more difficult to achieve, 633/200 = 3nm.
Tilt or Wedge
End mirrors that are not parallel cause a change in the phase of the beam across the etalon. This results in reduced finesse since not all the beam is emerging 'in phase' creating a bright fringe or 'out of phase' creating a dark fringe. The result is a low contrast mixing of dark and bright fringes.
Spherical Error
The same is true of spherical error, the phase of the light varies over the surface of the etalon due to the curvature of the surfaces. The errors caused by both Wedge and Sphere are predictable and can be calculated in specific ways which is why they are called out specifically in our calculations for expected performance.
Scatter and Material Losses
Scatter causes light to leak out of the etalon while materials will absorb light during each pass. These losses are usually insignificant if the proper coatings and materials are used unless the finesse is very high (above ~200).
Each of the parameters that effects the finesse also limits the finesse to some value. etalon-calc-small.jpg A mirror reflectivity of 60% limits the finesse of an etalon to about 6. In the same way all of the various defects each limits the finesse to some level and these limits can each be explored using our etalon calculator by hovering over the 'lambda' on the finesse label as shown on the right.
Specifying Etalons
Etalons are generally specified by
- FSR
- Finesse
- Transmission
- Wavelength range
- Clear aperture
All of these specifications are measurable, functional specifications. The reason that mirror reflectivity is not generally specified is that the reflectivity does not guarantee performance. Scatter, surface errors and tilt can significantly reduce finesse and transmission below the finesse that would be expected from a given reflectivity.