Measuring the FWHM, FSR, delta CWL, and cavity gap of Fabry-Perot etalons:

using the Fabry-Perot as an interferometer 


(Copyright François Rouvière  - Christian Viladrich)



Principle and optical setup

When a Ha etalon is illuminated by diffuse light coming from a hydrogen lamp (emitting mainly Ha), a system of interference rings forms at the infinite. This interference system can be imaged with a camera having its lens focused at the infinite (and also seen directly by eye without a camera lens).

-          The FWHM, FSR and gap of the etalon are derived from the measurement of the interference ring system (diameter and FWHM of each fringe).

-          The CWL is derived from the measurement of the diameter of the central ring (if any is visible) or better on a linear regression based on the diameters of the inteference rings.

The tilt of the etalon with respect to the light source has no impact, since the etalon is illuminated by diffuse light.

The aperture of the etalon sampled depends on the aperture of the camera lens.




Part 1 : Theoritical ground

François Rouvières (mathematician and co-author of Solar Astronomy) worked out the theoretical approach and derived the associated formulas. The full demonstrations are provided in the following documents:

a) Calculation of FWHM, FSR and cavity gap
FWHM .pdf
 Figures.pdf

b) Calculation of the delta CWL at normal incidence
Delta CWL.pdf

Part 2 : Implementation and detailed measuring procedure
1) Acquisition tips

-          Discharge lamp:

o   I use an Edmund Optics hydrogen discharge lamp. Probably all makes are equivalent.
o   The lamp emits hydrogen lines other than Ha. They are too faint to be a bother as long as a ERF is used. Otherwise a red filter (such as W25) can be used to block these lines.
o   Outgassing in the vacuum tube occurs from the electrodes. The tube progressively becomes progressively whitish due to additional emission lines. Out gassing does not occur when the tube is cold. It is therefore recommended to run the tube for a maximum of 30 seconds and allow it to cool before starting a new cycle.  In worst case, just use an ERF filter to cut the unwanted lines.

-          The diffuser:

o   To be placed about 20 cm from the hydrogen lamp.
o   Should be diffuse enough; otherwise the interference rings are not complete or not homogenous.

-          General light condition:

  • Measurements should be made in the dark. Otherwise, there is a continuum of brightness between the transmission peaks, which spoils the accuracy of the curve-fitting.  The camera lens:
        o   Should be focused to the infinity. Use manual focus and zoom on the image to check focus (possible with Nikon Z7 electronic viewer). Since focusing on the fringes might be difficult, it is helpful to focus the camera on a source at the infinite.
o   The quality of the optics and the accuracy of focusing determine the accuracy of the measurements. The lens should be of excellent quality even at full aperture. Otherwise the FHWM will be overestimated.
o   The focal length of the lens should be measured with 2-digit accuracy. This is easily done with Astrometry.net. For example, the focal lengths of the Nikkor 85 mm f/1.8S and the Nikkor 50 mm f/1.8S were respectively measured at 82.9 mm and 51.9 mm.
o   The aperture of the lens should be large enough to cover the aperture of the etalon.o   The lens should be close to the etalon in order to reduce vigneting and see as many fringes as possible.
    • The use of a tripod with an X-Y platform is quite useful, as well as a Z platform, in order to align the etalon and the camera lens. If the alignment is not good enough, the brightness of the fringe is not uniform, and/or more fringes are visible on one side of the field.

-          Camera:

o   To improve S/N and maximize dynamic range, it is recommended to use 100 ISO and 14-bit RAW mode.
o   Typical exposure times range from 1/4s to 5 s.
o   No distortion correction or diffraction correction applied.
o   To avoid vibration, the camera is released with a remote control.
o   Turn off lens/camera stabilization
o   Use manual exposure. At first, Autobracketing from -5 EV to +5 EV at 1 EV intervals is useful. However, if conditions do not change from one etalon to the other, exposure time should be pretty the identical for the same type of etalons.


2) Extraction of the intensity profile

Notes:
- If the 14-bit RAW files can't be read directly by the processing software, they can be converted to 16-bit files using Adobe DNG converter.
- The profile of the fringes in red channel is analysed along the horizontal diameter (X axis) or any other slice passing through the center of the fringe system.
 - If the S/N of the profile is low (i.e. noisy):
        - A 20-pixel 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 the S/N.
        - For example :
 
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.
Cropping a slice of fringes along the horizontal  diameter of the etalon. Note the central dip due to the CWL (red) offset at normal incidence.

-          - Export the intensity profile along the horizontal line passing though the center of the interference fringe system in .txt file (or other format).

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.

3) Overview of the measurement of the fringe system

The principle is to measure the position (X) and the FWHM of each fringe.
For this, a Voigt function (+ constant) is curved-fitted to each fringe. The constant is used to account for the offset of the camera. The result of the curve fit is the estimation of the position of each peak (X) and of the associated FWHM.
To do this, I use Fityk freeware:

  https://fityk.nieto.pl

- Measured data can be exported (menu Functions/Export Peaks Parameters), to be used for calculation.
- Some part of the work with Fityk can be probably scripted (see Fityk website).

o   The CWL of the etalon is calculated based on the radius of the central ring (if any): 

d CWL = i^2/n^2, where d CWL is in A, i in degrees, n is the index of the gap (1 for air, about 1.6 for mica).

The radius of this central ring is not measured by curve-fitting (its profile is not a Voigt), but just by measuring the X position of the peak. This is accurate enough in general. A more accurate method is described latter on.


Notes:
- If the etalon is reasonably good, the  fringe profile should be a Lorenztian function.
- If the etalon is not uniform etalon an/or specially bad, the fringe profile may be a Voigt function, or even worse a non symetrical Lorentzian or Voigt function.

4) Calculation of the CWL, FWHM, FSR and air-gap using the spread sheet


Fitting a Lorentzian + constant function to fringe #1 Fityk. The FWHM (9.2 pixel) and center (X= 73.6 pixel)  of the Lorentzian is extracted.



6) Calculation of the CWL, FWHM, FSR and air-gap using the spread sheet
 The spread sheet implements the calculations established by François Rouvières.

o   Input data (red color on yellow cell):-

- X position and FWHM of each fringe, left and right (measuring five different fringes is good enough).
- X position, left and right of the central ring (for calculation of the CWL).
-  Pixel size (microns) and focal length of the camera lens (mm): for calculation of the image scale in degree/pixel.
-  Cavity index (n=1 for air).
-  Wavelength of Ha (A).

o    Output data:

-  FWHM, CWL, FSR, gap.

The following extract gives an example of calculation.

(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 consistency of input data,
- 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 should be very close to constant (this is a way to check the accuracy of the measurements) => take average value over 3 couple of fringes => result is used to calculate the FSR in Angstrom : 8.5 A.

(2) 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 should be very close to constant => take average value over 4 fringes => result is used to calculate the finesse (=17.55), and FWHM = FSR/ Finesse.
Example of spreadsheet.

Consistency checks and first look at etalon non-uniformity:
- A difference between FWHM L and FWHM R could be an indication of etalon non-uniformity or wrong setup (poor lens quality).
- Another indication of non uniformity of an etalon is a bad curve-fit: non symmetrical profile.
- The column "calculated center" should show about the same value, otherwise there is something wrong with the measurements.
- The first column "delta cos 2" (0.00260, etc.) should give fairly similar results, otherwise something is wrong with the measurements.
- The second column "delta cos 2" (0.0000146, 0.0000148, etc.) is very sensitive to the accuracy of focus and optical quality of the lens. If some data do not seem accurate enough, they could be excluded from the calculation of the average FWHMA.
- However, it is better to measure four of five fringes. This allows a  better check of the consistency of the measurement between succesive fringes.

6) Consistency checks

- A difference between FWHM L and FWHM R could be an indication of etalon non-uniformity or wrong setup (poor lens quality).
- Another indication of non uniformity of an etalon is a bad curve-fit: non symmetrical profile.
- The column "calculated center" should show about the same value for all fringes, otherwise there is something wrong with the measurements.
- The first column "delta cos 2" (0.00260, etc.) should give fairly similar results, otherwise something is wrong with the measurements.
- The second column "delta cos 2" (0.0000146, 0.0000148, etc.) is very sensitive to the accuracy of focus and optical quality of the lens. If some data do not seem accurate enough, they could be excluded from the calculation of the average FWHMA.
- However, it is better to measure four of five fringes. This allows to better check the consistency of the measurement between fringes.


7) Correction to be applied to account for the FWHM of the hydrogen lamp

The method uses a hydrogen lamp as a light source. As the Ha lamp is not perfectly monochromatic, the measured FWHM is the convolution of the FWHM of the etalon and the FWHM of the hydrogen lamp.

            (FWHM measured)2 =(FWHM etalon)2 + (FWHM lamp)2
Given that the FWHM of the lamp was measured to be 0.263 A (see more info here), the relation between the measured FWHM and the actual FWHM is :




Comparison between measurements using the spectroscope and using the hydrogen lamp show reliability of results obtained with the hydrogen lamp down to 0.3 A etalons. However, and to be on the safe side, it is preferable to keep with the measured (and not deconvoluated) value when it is lower than 0.35 A.



Accurate measurement of delta CWL
In the method described above, the delta CWL (at normal incidence) is calculated on the basis of the diameter of the innermost diffraction ring. This can pose two practical challenges:

a) Given that the transmission profile is not symetrical, there is some incertainty in the measurement of the position of the transmssion peak. Furthermore, the profile it is not a Lorentz nor a Voigt, so curve fitting - though possible - requires a more elaborate reference function.
b) When the delta CWL is small, the "innermost ring" actually merges into the central inteference pattern, and no measurement is possible.

In order to get a more robust and accurate measurement of the delta CWL, it is better to use the following approach based on a linear regression on the diameter of the interference rings.

Theoritical grounds by François Rouvière:

delta_CWL.pdf

Formula (6) page 3 is used for the linear regression and calculate the delta CWL. Note that formula (5) is not accurate enough because the angles measured are not small enough.


Spread sheet example:
An example of calculation is provided in this spread sheet:

H-lamp-etalon-CWLb.xlsx

Calculation of the CWL is done in columns AC to AF.




NB : in the spreadsheet, two linear regressions are done:
- one including fringe #0,
- one excluding fringe #0.
As the diameter of fringe #0 is not measured by curve-fitting, but only by direct measurement of the position of its transmission peak, it can be argued that its measurement is less accurate than the other fringes. This would suggest not including this measurement in the regression.
However, making both calculations (with and without fringe #0) is probably informative of the level of accuracy of the measurement of the delta CWL.




Area of the etalon sampled
The area sampled (i.e. measured) on the etalon is determined by the diameter of the aperture stop of the camera lens (and also by vignetting due to the etalon barrel and lens combination).  For example, an 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 samples an area less than 10.6 mm.
- An easy way to understand this is as follows :
    - The fringe 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 determined by the size of the aperture stop of the camera lens (to an approximation, see bottom of the page).
    - In a perfect world, and in order to sample the entire surface of the etalon in a single shot and estimate the average FWHM and 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 because usually lens have a poorer optical quality at full aperture.
    - Otherwise, measurements made with a lens with a small aperture should be properly integrated  over the whole aperture of the etalon to estimate the FWHM at full aperture..
- 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  full-aperture FWHM: what is the most relevant?

The FWHM and CWL of an etalon may vary locally depending on the area sampled on the etalon.

(a) When the etalon is placed 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 performance 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 acrossf the full aperture of the etalon is correct only if it takes into account both the local values of FWHM and CWL.
- For example, let's consider an etalon whose local values of FWHM are all 0.3 A, but whose CWL changes dramatically of  +/1 A from one area sampled to the other. If we calculate the average FWHM over the full aperture of the etalon only from the local FWHM value statistics, then the result (0.3 A)  is wrong, because of the strong variation of CWL over the aperture. In fact, a correct calculation should include both FWHM and CWL statistics.
 
(b) For an etalon placed in the 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 cm.
- All  the area of the etalon within this central 2 cm diameter contribute equally to the contrast of the image.
- Accordingly, the relevant value for the observation is again the average (or integrated) values of  FWHM and CWL, and not the local values.



Accuracy of the measurement

The accuracy of measurement on :
- The quality of the lens used. The FWHM of the lens can be measured over the field of view and at various f-numbers using CCD inpector. See various examples here.
- The accuracy of the focal length of the lens. The focal length can be measured using Astrometry.net
- The accuracy of the focusing.
- The good sampling of the fringe profile in order to get a good estimate for the curve fitting. To be on the safe side, the FWHM of the fringe should be greater than 10 pixels.

Measuring air-spaced etalons is much easier than measuring mica-spaced etalons because:
- transmission is much higher (>60%) versus a few percent,  which means better S/N in fringe profiles because lower ISO can be used,
- smaller FSR (typically 10-12 A versus 20-30 A for mica-spaced etalons) and refractive index (1 instead of about 1.6), meaning a larger number of fringes can be measured (more than 5 versus 2 or 3 at most for mica-spaced etalons),
- the 2nd and 3rd interference fringes of mica-spaced etalons are very faint, so it is necessary to surexpose massively to detect them,
- for some reason (polarizers not properly aligned ?), it happends that the fringes of mica-spaced etalons tends to be less symetrical than the fringes of air-spaced etalons.


Additional note on the size of area actually sampled on the etalon due to 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. 

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