**Air-spaced and composite solid air-spaced F-P etalon theoritical performances**

**2) CWL shift and FWHM broadening in non telecentric lens systems**

**3) Daystar filter modelling and additional results**

**4) Air-spaced F-P etalon theoritical performances**

**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**

This part takes a look at the theoritical performances of air-spaced and composite solid air-space F-P etalon placed in front position (in front of the telescope aperture) and in a telecentric lens system.

The formulae of the F-P are from **Fabry-Perot math and bibliography.**

Information on solid air-spaced etalon are from US Patent 7142 573 B2 Nov 28, 2006.

Some comparisions are done with actual measurements made by Cyril Bazin and Serge Koutchmy of the Astrophysical Institute of Paris on Coronado F-P 40, 60and 90 mm (ref W11).

Most of the litterature available on Coronado filters assume these filters are air-spaced etalon. However, the situation is not that clear since some simulations made on this assumptions do not fit with available measurements or visual / imagin experience.

US Patent 7142 573 B2 Nov, 2006 shows the concept of a temperature compensated solid air-spaced etalon. It is not clear whether Coronado actually implemented this solution for any of their products.

Composite solid air-spaced etalons present the double advantage of increased acceptance angle and thermal stability of the CWL. The idea is to partially fill the cavity with a solid parallel plate filler of refractive index n'. The filler is optically contacted to the one of the etalon mirrors. The refractive index of the resulting F-P etalon is the weighted average of the index of the air and solid plates.

Accordingly, the following analysis will distinguish air-spaced and solid air-spaced etalon. As little if any information are available, we will assumed a 1.4 refractive index for the composite solid air-spaced etalon.

The following parameters are taken from [W11] measurements :

- Free Spectral Range = 9.57 A

- Finesse = 13.3 A

- FWHM= FSR / Finesse = 0.72 A

- Order of interference = 643

- No assumption is required on the refractive index of the space at this stager.

The modelling supposes that the CWL is equal to 6562.85 A at normal incidence, while measurement gave 6562.94 A.

The following figure compares the transmission curve of the modeled Coronado 0.7 A with the modeled 0.7 A mica-spaced F-P Daytar. The Daystar has a Free Spectral Range (= distance between peaks of tranmission) nearly twice as large as the Coronado:

__FWHM of the narrow band blocking filter :__

In order to select the peak of tranmission at 6563.85 A, a blocking filter blocks the undesired orders.

The following measurement for the BF15, courtesy from Peter Höbel (see additional measurements at http://www.sonnen-filter.de/) gives FWHM = 7.8 A.

A measurement for the BF30 (Bazin and Koutchmy ref W11) gives FWHM = 7.5 A with a 30% peak transmission.

These values are similar to the FWHM of the blocking filters of Daystar Ha filters (about 5 to 10 A FWHM).

The profile of the transmission curve of a one-cavity F-P is a Lorentz curve which means that the foot of this curve is rather large. In other terms the bandwidth measured at half maximum (FWHM) is only part of the story of the ability of the filter to select the chromosphere light (H alpha) and cut the photosphere light (every wavelenghts outside Ha). Obviously, the contrast between the chromosphere and the photosphere increases as the FWHM descreases, still ... there is a lot of "out band pass" light, ie. light from the photosphere that is transmitted by the F-P filter, because of this large foot (or "tail").

The profile is simular to a Daystar of 0.7 FWHM (not surprisingly ...)

The bandwidth at 10% transmission is three times wider than at 50% transmission. This is the classical value for one cavity F-P filter.

If we remember that the photophere is much brighter than the chromosphere, we can understand that the bandwidth at 10% (or even 5%) transmission will play an important role in the contrast of the Ha images.

One way to have a steeper profile is to double stack F-P filters. The FWHM of the double stacked Coronado is about 0.5 A, the transmission profile is steeper with a bandwidth at 10% equal to 2.3 FWHM (instead of 3 FWHM).

The following table gives the bandpass in function of the transmission level. These are the values for any perfect F-P etalon, air-spaced or not (Lorenztian transmission curve):

Transmission Bandpass

(simple stack F-P)

Bandpass

(double stack F-P of same FWHM)

50% FWHM FWHM b = 0.6436 * FWHM 10% 3.00 * FWHM 2.28 FWHM b = 1.47 * FWHM 5% 1% 9.95 * FWHM 3.00 * FWHM

E) CWL in function of the angle of incident light in a collimated beam :From now on, it is necessary to make an assumption on the refractive index of the spacer. We will assume two values : n= 1 (for air-spaced F-P etalons) and 1.4 (for composite solid air-spaced F-P etalons).

If we assume Coronado filters are air-spaced, then a 1° tilt shifts the CWL by 1 A (versus 0.4 A for mica-spaced F-P etalon).

If we assume they are composite solid air-spaced, then a 1 ° tilt shifts the CWL by only 0.6 A.

F) Jacquinot spot and sweet spot (collimated beam with field angle) :

The Jacquinot spot is defined to be the region over which the change in wavelenght does not exceed sqrt (2) times the etalon bandpass. We can derive two expressions of Jacquinot spot :

with p = order of interference

F = finesse

q : in radians

For a 0.7 A Coronado 60 mm place in front prosition, and assuming an air-spaced F-P, we have p = 643 and F = 13.3, which gives an angle q = 1°, which is consistent (not surprisingly) with the transmission curves given above.

How this "Jacquinot spot" fits with the "sweet spot" of visual solar observers ? If we take a closer look at the transmission curves, we can see that the transmission at Ha for a 1° tilt is only 12%. Accordingly, the image will be dominated by the photosphere contribution.

If we want to keep the transmission in Ha greater than 60%, we have to limit the tilt of the air-spaced of the F-P etalon to 0.5°. This means that if the CWL is tunned on Ha on one limb of the sun, then the opposite limb will be blue shifted by 0.25 A, with a transmission of only 60% transmisison for Ha. If we have a composite solid-air space F-P etalon with a 1.4 equivalent refractive index, then the sweet spot increases to 0.7°.

G) Comparison with actual measurements(Coronado 60 mm placed in front position)

CWL shift = f (tilt)The following measurements are courtesy from Cyril Bazin & Serge Koutchmy (see summary in W11). When the 4' light cone is considered in the calculation, the theory for an air-spaced F-P etalon comes in close agreement with the measurements.

The measurements for the Coronado 40 mm and 60 mm are essentially identical.

We can conclude that the Coronado 40, 60 and 90 mm are indeed air-spaced F-P.

Unfortunately, no PST has been yet measured.

H) FWHM in function of F/D ratioin a telecentric lens system

Comparison of mica-spaced, composit solid air-spaced and air-spaced F-P etalons :Here, we assume we have an incident convergent beam with no field angle. This is the situation when using a F-P etalon behind a telecentric system. The angle of the cone of light is set by the F/D ratio of the instrument.

Not surprisingly, air-spaced F-P (n= 1), composite solid air-spaced F-P(n= 1.4), or mica spaced F-P (n=1.6) gives very significant results.

The curve for the air-spaced etalon is in line with ref W3. Professionals used air-spaced F-P etalons in light beams with F/D > 100.

Conclusion for air-spaced F-P etalons :a) Air-spaced F-P etalons work nicely when placed in front position (in front of the objective of a refractor). The sweet spot is large enough to cover the full disk of the sun with a limited CWL shift (< 0.25 A).

The only problem is the price of the etalon when diameter increases ...

b) Placed in a telecentric beam at F/28, an air-spaced F-P etalon is a very poor performer. So, it is difficult to understand how a PST 1 A air-spaced F-P etalon can give any good results in a F/28 telecentric beam. Maybe there is something not properly considered properly in all this analysis ? Or could it be an indication of the PST etalon being actually a composit solid air-spaced etalon ? This would be surprising for a low cost F-P etalon.

A measurement of the CWL = f (tilt) for the PST could help solving this mystery...

c) Air-spaced F-P can also be used in collimated beam. An example is the 200 mm Zeiss refractor of Rogerio Marcon with a Coronado 60 mm mounted in a collimated beam :

http://www.astroimagem.com/Mybestimages/Solaractivity.htm

An another example is the "PST modification" analysed here :