Measuring the bandpass of a narrow band filter or an etalon with a slit spectrometer


Examples of configurations

Configuration with 125 mm f.l. collimator, 125 mm f.l. imager and ASI290.

Left: the filter in placed in front of the slit of the spectro. The Sun is the source of light, resulting is a f/115 nearly collimated beam.

Right: the filter is placed in a telecentric beam (f/30) on a Celestron 8. The spectro is mounted on the filter output.



Methodology

The procedure is as follows:

- Take an image of the solar spectrum without the filter to be measured. Image_1 = Solar_Spectrum_Image.

- Take an image of the solar spectrum transmitted by. Image_2 = Solar_Spectrum_With_Filter.

The transmission profile of the filter is given by

            T (wavelength) = Solar_Spectrum_With_Filter (wavelength) / Solar_Spectrum_Image (wavelength)

The transmission profile is then curve-fit using Fityk with the relevant function (typically Lorenztian function for a single-stack etalon) in order to estimate the FWHM.

Measurement can be done using:
- a nearly collimated beam, when the slit is directly illuminated by the Sun (equivalent f-115 ratio),
- the telescope optical setup (f-ratio, telecentric mount or not).


Example 1: measuring an air-spaced etalon in a collimated beam

Optical setup: Sun => etalon => Solex with 7 microns x 6 mm slit, fc = f i =200 mm, 2400 l/mm grating, ASI 290, 0.0471 A/pixel dispersion

Top image : spectrum transmitted by the etalon, linear vizualisation.
Middle image : same spectrum, but this time with log vizualisation is order to show the faint line (identified by a cross) which will be used to register this spectrum with the solar spectrum.
Bottom image : solar spectrum (without etalon), with the identification of the line (see cross) which will be used to register this spectrum with spectrum transmitted by the etalon. .




1) The profile of each image along the X-axis is extracted (if the wavelength axis is along the X axis).
If the images are noisy, it is possible to take the average over a few pixels (e.g. 15 pixels) along the Y axis.

2) Registration of both X profiles along the X axis (wavelength).
Due to the differential  mechanical flexure that occurs between the image of the solar spectrum and the image taken with the etalon, the two spectra are slightly offset along the X axis.
Registration is done by identifying the X position of the same spectral line in both spectra (X1 and X2). In this example, the line used is identified in both spectra by a cross. The X1 and X2 positions are measured by placing the cursor on the line (AstroArt software in this case, but any  software giving the (X,Y) position of the cursor will do).
In this example, the X offset (=X2-X1) was found to be equal to 30 pixels (= 1.4 A)
So, the profile of the spectrum transmitted by the etalon is shifted in EXCEL according to the measured offset (in pixels).

NB :
- The registration can be fine tuned using two lines instead of only one.
- At this stage, there is no need to know the relation between the X axis and wavelengths.

3) Calculation of the filter transmission profile:     T(wavelength)  = X-profile-of-spectrum-transmitted-by-filter(wavelength) / X-profile-of-solar-spectrum(wavelength)
This can be done in EXCEL.
No need to know the relation between X axis and wavelengths.

4) Curve fit of this profile using a Lorentzian function (+ constant) using Fityk https://fityk.nieto.pl/
- green line = recorded data,
- yellon line = curve-fit.



NB : in the above figure, the unit of the X-axis is in wavelength (A), but it could also be left in pixels. In the later case, we go from the FWHM in pixels to the FWHM in Angstroms by multiplying the former by the dispersion of the spectroscope (A/pixel).


5) Check whether curve-fit is good.
If not, try curve fitting each peak separately. If still bad, try the Voigt function. If still bad, try SpiltLorentzian, then SplitVoigt.
A bad curve fit may be an indication of an incorrect measurement protocol, or a very bad etalon.

6) Plot the measured FWHM and FSR :


FSR increases with the order of interference as expected.
The FWHM is measured at the fringe closest to Ha. Taking the average of the two neighboring fringes is also a good option since the Ha fringe measurement may be slightly  distorted because of the non linearity of the sensor.

Note for air-spaced etalons :
It is also possible to work only with the image of the spectrum transmitted by the etalon. This avoids the trouble of registering both spectra (spectrum only with Sun, spectrum with the filter).
It that case, the Ha fringe is not to be consisdered in the measurement (because its profile is spoiled by the Ha line).
The accuracy of the measurement is lower.


Example 2: measuring a mica-spaced etalon in a telecentric beam

Measuring tips and accuracy of the measurement

Equipement:

- Accurate focusing of both the collimating and the imaging lenses is very important, especially in the near UV (Ca, K, or H). Note that the focus for Ca K is different from the focus for Ca H, which is different from the focus for Ha.

- Diffraction limited collimating and imaging lenses are essential, especially in the near UV where lens spherical aberration is usually very important.

- To maintain accurate focusing throughout the measurement, it is important that the temperature of the spectroscope be as stable as possible. Note that the CTE of aluminum is twice that of PETG.

- It is advantageous to cover the spectroscope with a black shroud to minimize background light. Check light leakage using long exposure times and blocking light on the slit.

- For acquisition : 

        - use maximum number of bits of the camera,

        - use the minimum value gain, 

        - set the offset so that there is no threshold effect, 

        - chose an exposure time long enough to fill the histogram up to 80-90%. Note that exposure time can goes up to 1s for very dark mica-spaced etalon.

        - choose an acquisition duration long enough to have a good S/N. Typical duration ranges from 10 s (for high transmission air-spaced etalons) to more than 30 s (for low transmission mica-spaced etalons). 

        - no not change any acqusition parameters  when recording the solar spectrum and the spectrum, except the exposure time.

        - stack all the frames of the SER file (no registration, no selection of images).

- Non linearity of the sensor can  cause some distortion in the estimation of the transmission profile of the filter. The linearity of the sensor can be checked using SharpCap. See some examples here : Sensor linearity

- Tuning etalons :

        - It is better to tune etalons away from Ha for the measurement. Indeed, setting the etalon right on Ha can cause distortion effect due to non-linearity of the sensor. 

        - It is better to measure air-spaced etalons without their Blocking Filters because more fringes can be measured.

Measurement:

- For etalons, the curve fit should be done with a Lorentzian curve (or a Voigt if the etalon is not uniform). A fit with a Gauss function is not accurate enough.

- As a rule of thumb, in order to measure a filter with a bandpass equal to FWHM, the dispersion of the spectroscope should be greater than FWHM/4 (e.g. 0.125 A/pixel dispersion for measuring a 0.6 A etalon). Otherwise, the curve fit will not be accurate enough.

- For the same reason, and more importantly, the spectral resolution (= bandwidth of the spectroscope) should be related to the FWHM of the etalon being measured. A spectral resolution better than FWHM/2 is recommended.

Cross-check results:
In general, it is recommended to cross-check the results with different approaches (spectroscopic approach in collimated and telecentric beam, and the hydrogen lamp).

Note on measuring the dispersion of the spectroscope

To perform the previous measurements, we need to know the dispersion of the spectroscope (A/pixel). It can be measured very easily, without the need of using reference lamps or a specialized software dedicated to spectroscope.

The dispersion of a spectroscope depends on the wavelength. The good news is that we don't need the full relation dispersion = f (wavelength). We only need to know the dispersion around the Ha line (or any other line of interest to measure a narrow band filter). 

Two methods can be used :

(1) - Simply  use the Simspec.xls spreadsheet provided by Ken Harrison. 

The main uncertainty in this approach is that the focal length of the collimating and imaging lenses is known to an accuracy of  +/1%for the designed wavelength (for lenses provided by Thorlabs), and changes with wavelength.

Still, resulting accuracy is in the range of +/-5% which is quite reasonable for the use we have.

(2) - Actually measure the dispersion around Ha using the solar reference spectrum provided by Bass2000. 

https://bass2000.obspm.fr/solar_spect.php

Using this reference very high resolution reference spectrum, the wavelength of the solar lines can be measured to a accuracy of better than 0.01 A, which is much is more than enough. This work has already been done here:



What remains to be done  is to measure the X positions of these lines with your favorite processing software (using the cursor).  

We the have : dispersion = delta (wavelength) / delta ( X)

By measuring  several pairs  of lines, we can check if the results are consistent, and take the average value and estimate the uncertainties on the measurement.

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