2) Measurement procedure
- The 14-bit RAW file is converted in 16-bit file with Adobe DNG converter.
- The profile of the fringes in red channel is analysed along the horizontal diameter (X axis).
- To increase S/N :
- A 20 pixels high slice is cropped from the image.
- In this cropped image, the curvature of the fringes is negligeable,
allowing to average the columns (Y axis) in order to increase S/N.
Coronado SM III 60 mm (tilt) - Nikon Z6 with 85 mm S f/1.8 lens at f/1.8 - 100 ISO - 1/2s - 14 bit acquisition - RAW mode - Red channel.
a slice of fringes along the horizontal diameter of the etalon.
Note the central dip due to the CWL (red) offset at normal
- The X profil of the cropped image is extracted.
SM III 60 mm (tilt) - Nikon 85 mm f/1.8 S at f/1.8 - 100 ISO - 1/2s.
X profile along the 12 fringes central fringes.
The profile and position of each fringe is measured with Fityk :
- The Airy function defining the F-P interferometer
transmission with the incidence angle can be approximated by
a Lorenztian function at each fringe.
- Accordingly, each fringe is least-squared fitted to a Lorentzian
+ linear function. The linear function is intended to model the
- The FWHM and the center of the Lorentzian function are estimated by Fityk.
- The CWL offset of the etalon is calculated based on the radius of the central ring (if any) : delta CWL = i^2 / n^2, with delta CWL in A, i in degrees, n is the index of the gap (1 for air, about 1.6 for mica)..
- The FWHM and center of the other fringes are used to calculate the FSR and FWHM according to the formulae presented in part 1.
a Lorentzian + constant function to fringe #1 Fityk. The
FWHM (9.2 pixel) and center (X= 73.6 pixel) of the Lorentzian is
when the etalon is of average quality, the transmisison profile
can depart signficantly from a Lorenztian fonction. In these
cases, a Voigt function might be more relevant, or a split Lorentz, or
even a split Voigt (in very bad cases).
3) Calculation spread sheet
The following extract gives an example of calculation. according to the formulae provided in the theoritcal part.
(1) On the left side, calculation of the FSR:
- red color in yellow cells: measurements of FWHM and position of 6 fringes (left) and 5 fringes (right)
- calculation of fringe centers: used for checking
- for each fringe: radius of the fringe = 1/2 diameter
- for each fringe, calculation of the radius of the fringe i in radian (conversion pixel -> micron -> radian)
- for each fringe, calculation of cos^2 (i) (see explanations in the pdf)
- for each couple of sucessive fringes n,
n+1, calculation of cos^2 (i) fringe (n+1) - cos^2(i) fringe (n)
=> this value is a constant (if measurements are accurate) =>
take average value over 3 couple of fringes => result is used
to calculate the FSR in Angstrom : 8.5 A.
On the right side, calculation of finesse and FWHM:
- for each fringe, calculation of the average value of the FWHM of the fringe in pixel (left/ right), conversion in radian,
- for each fringe, calculation of angles i+k, i-k (see explanations in the pdf).
- for each fringe, calculation of FWHMA in radian^2,
- for each fringe, calculation of delta cos^2 (see
explanations in the pdf) => this value is a constant => take
average value over 4 fringes => result is used to calculate the
finesse (=17.55), and FWHM = FSR/ Finesse.
Some easy checks:
- A difference between FWHM L and FWHM R could give some
indication of etalon non uniformity, or wrong setup (poor lens quality).
- The column "calculated center" should display about the same value, otherwise there is something wrong with the measurements.
- The first column "delta cos 2" (0.00260, etc.) should give pretty
similar results, otherwise there is something wrong with the
- The second column "delta cos 2" (0.0000146, 0.0000148, etc.) is very
sensible to the accuracy of focus and optical quality of the lens.
If some data appear not accurate enough, they could be excluded from
the calculation of the average FWHMA.
4) Area of the etalon sampled
The area sampled (i.e. mesured) on the etalon is set
by the diameter of the aperture stop of the
camera lens (and also by vigneting due to the
etalon barrel and lens comnination). For example, a 85 mm f/1.8 lens samples an area
of (approximately) 46.1 mm in diameter on the etalon, while the same lens at f/8 sample an area less than 10.6 mm.
- An easy way to understand this is as follows :
- The fringes system is at the infinite.
- The size of each fringe on the camera sensor depends on
its angular diameter and on the focal length of the lens (just like the
size of any celestial object).
- The light beam coming from any point of a given fringe
is a collimated beam intercepting the whole surface of the etalon
(except vigneting due to the etalon barrel).
- Accordingly, the size of the area sampled on the
etalon is deterrmined by the size of the aperture stop of the camera
lens (to some approximation, see bottom of the page).
- In a perfect world, and in order to sample the
surface of the etalon in a single shot and estimate the average FWHM
FSR over the full aperture of the etalon, we should use a lens whose
aperture is similar to the aperture of the etalon. This is an issue
since usually lens have a poorer optical quality at
- Otherwise, measurements made with a lens with a
small aperture should be integrated properly over the whole
aperture of the etalon.
- The farther the camera (or the eye) from the etalon the lower
the number of fringes is seen (the angular diameter of the fringes is
unchanged, while the angular diameter of the etalon and central
spacer decreases). This does not change thes size of the area sampled
on the etalon.
Local or integrated FWHM: what is the most relevant?
The FWHM and CWL of a filter vary locallly depending on the area sampled on the etalon.
(a) If the etalon is placed in front position in front of the aperture of the telescope:
- All parts of the etalon contribute equally to the quality (i.e.
contrast) of the image. Accordingly, the relevant value to qualify the
etalon performence is the FWHM (and CWL) integrated over the full
aperture of the etalon, and not the local values of FWHM and CWL.
- However, the mapping of the local values of FWHM and CWL can still be
used to calculate the average (or integrated) values over the full
aperture of the etalon. The integration of the FWHM of the full
aperture of the etalon is correct only if it takes in consideration both the local values of FWHM and CWL
- For example, let's assume an etalon with local values of FWHM
all equal to 0.3 A, but whose CWL changes dramatically of +/1 A
for one area sampled to the other. If we compute the average FWHM over
the full aperture of the etalon only from the local FWHM value
statistic, then the result (0.3 A) is wrong, because of
the strong variation of CWL over the aperture. Indeed, a correct
calculation should taken into account both FWHM and CWL
(b) For an etalon placed in rear position:
- Let's assume an observation of the Sun with a 2 m focal length
telescope. The diameter of the solar disk at the focal plane is about 2
- All the area of the etalon within this central 2 cm diameter contribute equally to the contrast of the image.
- Accordingly, ther relevant value for the observation is again the average
(or integrated) values of FWHM and CWL, and not the local ones.
More on the size of area sampled and camera lens vigneting effect :
These ray-tracing for Nikon 50 mm f/1.8 are from : https://www.photonstophotos.net
Nikon 50 mm f/1.8 at f/1.8 - The entrance pupil (Pd in blue) is 27.78 mm.
Nikon 50 mm f/1.8 at f/4 - The entrance pupil (Pd in blue) is 12.42 mm.