Contrast factor of a stack of F-P etalons and blocking filter

1) Center Wave Length (CWL) and bandpass (FWHM) as a function of f-ratio of the telescope and tilt of the F-P filter (in collimated beam, telecentric beam or non-optimized telecentric system)

2) CWL shift and FWHM broadening in non telecentric systems (refractors and Cassegrain telescopes) and useful formulae

3) Daystar filter modelling and additional results

6) Contrast factor of the F-P etalon and blocking filter assembly

7) Contrast factor of the F-P etalon : test of various stacking schemes

8) Fabry-Perot math and bibliography

### Selectivity of a filter combination

The following figure compares the intensity of the solar spectrum and the transmission profile of a 0.6 A FWHM F-P etalon :

The width of the Ha absorption line is about 1 A at 4000 normalized intensity, 0.8 A at 3000 intensity and 0.55 A at 2000 intensity.

The role of the Ha filter is to block the unwanted light of the photosphere, in other words the light outside this absorption line (strickly speaking there is still some amount of photosphere light even in the Ha line ...).

We already know that some amount of light is transmitted by the F-P etalon outside its own FWHM. A more interesting question is how much light is transmitted by the filter outside the Ha line ? The ideal filter combination would block everything out of the Ha line.

Based on the solar spectra detailed profile (BASS2000) and on the transmission profile of the filter combination, we can calculated :

- The amount of light transmitted by the filter combination inside the Ha line. This is the "on-band light".

- The amount of light transmitted by the filter combination outside the Ha line. This is the unwanted "off-band light".

These values are calculated assuming a 1 A width for the Ha line (ie. 4000 intensity level). This means that the 1 A window centered on Ha is declared "on-band", while the outside of this band is declared "off-band".

NB : the conclusions derived bellow would be about the same if we considered a 0.8 A width for the Ha line).

Let's call "selectivity" of the filter the following ratio:

selectivity = on-band light / (total light transmitted by the filter)

This ratio is a very good indicator of the blocking capacity of the filter, and its ability to cut the unwanted photospheric light outside the Ha line.

- Selectivity is equal to 100% when the filter transmits the light from inside the Ha line, but no light outside Ha (square transmission profile centered on Ha).

- If slectivity is equal to 50%, it means that the light coming from inside the Ha line accounts for half of the total amount of light transmitted by the filter.

Modelling of the blocking filter (BF):

In order to calculate the selectivity of a filter stack, we need to take into account the Blocking Filter (BF). Indeed, a F-P etalon is not used alone but associated with a blocking filter whose role is to cut the unwanted interference orders of the F-P etalon.

The tranmission profile of the BF is modeled on the following basis:

- DayStar appears to use an Andover 10 A 2-cavity filter (ref 656FS02). The profile for the "unblocked" version is available from Andover at 0.2 A resolution (txt file). It is taken as it is, with its CWL at 6563 A (+2 A/-0) and peak transmission of 72%. The "blocked" version would have a peak transmission of 45% (Andover data).

- SolarSpectrum uses a 3-cavity 10 A filter. Its profile is approximated by the one of Alluxa 656.3-1_OD4_7018 at 0.4 A resolution.

The following figure compares the profiles of:
- an Andover 10 A 2-cavity BF,
- an Alluxa 10 A 3-cavity BF,
- a 1.0 A F-P mica-spaced etalon (assuming a FSR of 26.8 A, from actual measurements),

For comparison, the FWHM of  the BF associated to the Coronado air-spaced etalon lies in the range of 6.5 A to 7.8 A (sources: K.M. Harrison, Bazin & Koutchmy, P. Höbel, C. Viladrich)
The following figure compares:
- the profile of a 0.7 A F-P air-spaced etalon (assuming a FSR =  9.3 A, from actual measurements).

Ha selectivity of various filter combinations in collimated beam

The following table presents the Ha selectivity of various filter combinations, based on the previous data.

Etalons #1, #2 and #3 are a mica-spaced etalons (FSR = 26.8A).

- With a single stack 0.7 A etalon, only 35% of the transmited ligt comes from the chromosphere.

- While the FWHM of the 0.6 A + 1.5 A combination (FWHM = 0.53 A) is much larger than the FWHM of a single stack 0.3 A filter, its selectivity is about the same (53% versus 57%). In other words, the blocking of the photospheric light is about the same in both cases. It is also to be noticed that the width at 1% tranmission is about the same (2.8A versus 2.9 A).

- The selectivity of the 0.6A + 1.0 A combination is greater than the selectivity of a single 0.3A etalon, which is consistent with imaging tests (see next page).

- The classic 0.7A + 0.7A stack has twice the selectivity of the single stack 0.7A : 75% of the light transmitted is from the chromosphere, versus 35% in the 0.7A single stack. This is the very reason of the improved contrast. The selectivity is even much better than the one of a single stack 0.3 A (75% versus 57%), in line with a much narrower width at 1% transmission (2.1 A versus 2.9A).

- The use of a 2-cavity Andover BF or a 3-cavity Alluxa BFdoes not change the selectivity because of the large FSR of mica-spaced etalon. The main advantages of the Alluxa BF are improved peak transmission, flat top transmisision, and much longer time life (hard-coating instead of soft-coating).

- In case of double-stack combinations, the selectivity is about the same when etalon #2 is an air-spaced etalons (FSR =  9.3 A) instead of a mica-spaced etalon.

Ha selectivity of various filter combinations in a f/28 telecentric beam (mica-spaced etalons)

The selectivity decreases compared to the case in collimated beam.

NB : Etalons 1, 2 and 3 are mica-spaced etalons except when noted.

Ha selectivity of various filter combinations in a f/28 telecentric beam - Comparison of mica-spaced and air-spaced etalons

Some imaging tests

FP-contrast-2.htm

c) Visual tests with a small 55 mm diameter fluorite refractor and Baader TZ4 :

0.6A etalon (rear position) : filaments are faint, strong double limb effect.

0.6A etalon (rear postion) + 0.9 A etalon (rear position) : filaments are darker, faint double link effect.

0.7A etalon (front position) + 0.6A etalon (rear position) : filament are dark, no double limb effect.

=> the photosphere and the double limb effect are is no longer visible with the 0.7 A + 0.6 A stack

Example of transmission profiles for double-stack combination (mica-spaced etalons)

### Conclusions on contrast in filter stacking

1) The FWHM is not the relevant indicator to assess the Ha selectivity of etalons (single or double stack).

2) The width at 10% or 1% peak transmission is a more relevant indicator of the selectvity of a combination (i.e. of the ratio between the light transmitted by the filter from the chromosphere to the total transmitted light). It is a good indicator for comparing and ranking single stack and multiple stack combinations.

3) A double stack 0.7A combination is more selective than a single stack 0.3A filter (75% selectivity versus 57%).

4) A double stack 0.6A and 1.5A combination gives about the same selectivity as a 0.3 A single stack etalon. This is confirmed by actual tests.