**Analysis of mica spaced Ha Fabry-Perot etalons **

**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**

Let's start with the simplest setup : a collimated beam (ie. a set of parallel beams of light) is incident on the F-P at an angle from normal incidence :

1) Center Wavelength Shift with the incident angle :In these conditions, the Center WaveLength (CWL) of the F-P filter is shifted to the blue when the incident angle increases. For small incidence angles we have

(1):Here are the simulations for a mica-spaced F-P such as Daystar filters:

The calculated curve matches perfectly the curve given in Daystar white book for Helium D3 line (5873 A).

The blue shift of the center wavelength with the tilt is not a big deal for thermo-regulated filters, since the CWL can be tuned by increasing the temperature of the filter oven (increasing the temperature of the filter shifts the CWL to the red).

2) FWHM as a function of the tilt :In a collimated beam, the change of FWHM with the incident angle is completely negligable

(2):A a numerical example, a FWHM 0.6000 A filter would become a FWHM 0.6001 when tilted by 1°. Not a big deal ...

## B) CWL shift and FWHM broadening in a convergent beam and with a field angle (general formulation)

Now we have seen that the result of tilting a F-P in a collimated beam is to shift the CWL to the blue (with no change in the FWHM), let's be ambitious and see what is hapening to the CWL and the FWHM in the general case of a converging beam falling on an etalon at a field angle.

Here are the three different possible cases depending on the optical setup :

And the associated formulae for the CWL shift and FWHM broadening **(3)**

Let's try to decipher these cases and formulae.

First, the notations :

- the incident beam of light is a converging cone of light whose half angle is set by the F/D ratio : q = D / 2F.

- the axis of this cone of light strikes the F-P at an angle c from normal incidence. This angle is called the "field angle".

Now, let's have a closer look at our three cases :

- Case 1 : If we use a telecentric system, then the field angle is 0 all over the field of view. This is nice since the CWL shift and the FWHM broadening will be minimal (they are set by the F/D ratio of the instrument), and constant all over the field of view. Obviously, this is the optimal situation when using a F-P at the focus of instrument. The CWL blue shift is equal to half the D FWHM.

We can notice that in a non telecentric system the field angle is equal to 0 only on the optical axis.

- Case 2 and 3 : If we don't use a telecentric system (or if the telecentric system is not optimised for the focal lenght of the telescope), then we have a field angle, which could be lower (case 2) or greater (case 3) than the cone half angle.

Let's have a look at these different situations in order to understand what are the implacations.

Let's start with the optimal case which is a F-P etalon in a telecentric lens system.

In a telecentric system, the field angle is equal to 0. In other words, the axis of the all incident cones of light are perpendicular to the F-P (figure taken from Zeemax help) :

See also a full drawing of a telecentric system here : http://www.pbase.com/p_zetner/image/151958178

This means that the CWL shift and FWHM broadening will be the same all over the field.

**CWL shift :**

The angle of the incident rays inside the falling cones of light ranges between 0 to theta. Each incident ray builds up its own associated shift of CWL. Accordingly, the CWL associated to the full cone of light is the integration (or average) of all these CWL over all the angles within the incident cone.

We can make a guess of this average CWL. If we notice that there is as much light falling in the cone outside and inside a solid angle of sqrt (2) x q, we can guess that the average CWL due to the cone light is equal to the CWL shift associated with an incident ray at an angle of sqrt (2) x q, or half the CWL associated to q in a collimated beam. This intuitive guess is confirmed by the full maths of F-P etalon (see ref W10 page 288- 291).

This CWL is not a big deal in a telecentric system as it is uniform all over the field of view an can be conpensated by an increase of the temperature of the F-P etalon.

__FWHM versus focal ratio__

It is well known that narrow band air-spaced or mica-spaced F-P etalon should be used with a high focal ratio. Daystar recommends a F/D 30 ratio for their Ha mica-spaced F-B etalons. However, it is interesting to investigate the dependency of the effective FWHM on the focal ratio.

An other way to present the formulae given in **(3)** is** (4)** :

Mica (muscovite) being birefringent, the refractive index depends on the polarisation of the light. The refractive index ranges between 1.59 to 1.64 (na = 1.55 to 1.58, nb =1.58 to 1.61). Measurements at Meudon solar tower (see ref W9) on a 0.35 A Daystar filter were consistent with n = 1.6173. If we adopt this value, we can simulate the evolution of the FWHM of Ha Daystar filters according to the F/D ratio:

Nominal FWHM | Effective FWHM at F/30 | FWHM broadening |
---|---|---|

0.3 A |
0.46 A |
+0.16A |

0.4 A |
0.53 A |
+ 0.13 A |

0.5 A |
0.61 A |
+ 0.11 A |

0.6 A |
0.69 A |
+ 0.09 A |

Obviously, the broadening of the FWHM for a given F/D ratio is more detrimental for narrower bandpass filters.

Daystar white book gives a very different curve for a 0.5 A Ha filter (see the blue curve at left herafter). It is not clear whether Daystar's curve comes from actual measurements or calculations. However, the only way to get close to this curve, is to assume a value of 1.96 for the refractive index of the spacer (red curve at right). This would be very strange since this value is far away from the actual refractive index of mica, and from the measurements made on the 0.3 A Ha Daystar :

**Telecentric lens system and tilted F-P etalon :**

In real life, the F-P etalon might be tilted for at least two reasons. :

- angular slop of poor focusing mechanism. Daystar says that most of Crayford focusers have a small 0.5° angular slop(see http://www.daystarfilters.com/Quark/QuarkUniformity.shtml)

- we do want to tilt the F-P etalon (typical + or -1°) in order to quickly change the CWL (typicaly + or - 0.5 A). Indeed, tuning the CWL by increasing of decreasing the temperature of the F-P takes 10 mn or so.

Strictly speaking, we are no more telecentric in these conditions since we have a constant field angle all over the field. This field angle shifts the CWL (we have seen this is not an issue) and broadens the FWHM uniformly over the field (which reduces the contrast).

**FWHM in function of F/D ratio and field angle for mica-spaced F-P etalon :**

We can see that the tilt angle has a huge impact on the mica-spaced F-P performance. This impact is even larger for an air-spaced F-P etalon.

__A pratical maximum allowance of 0.1° tilt can be considered if we want to keep the FWHM broadening to lower than 0.05 A .__

A telecentric lens system is designed for an objective (or a mirror) of a given focal length. In other words, if we use a telecentric system optimised for a refractor of 800 mm FL on a refractor of 2000 mm FL, we get a degradation of the etalon performance. This is because the field angle is no longer equal to zero all over the field of view as in a true telecentric system, but varies according to the linear distance from the optical axis. The following formulae are from Gene A. Baraff (ref W1) **(5)** :

Given the formulae** (3)** and** (5)**, we can calculate the CWL shift and FWHM broadening resulting from the use of a non-optimized telecentric system .

For illustration, we will take the example of the Baader TZ2 (X2 magnification factor) and TZ4 (X2 magnification factor) telecentric systems, both optimized for 800 mm focal length, and we will check out their performances on three refractors of 440 mm, 1100 mm and 2050 mm FL :

- (a) A 55 mm F/8 refractor of 440 mm focal length. The effective focal ratio with the TZ4 is 8 x 4 = 32.0.

- (b) A 150 mm F/7.3 refractor of 1100 mm focal length. The effective focal ratio with the TZ4 is 7.3 x 4 = 29.2.

- (c) A 230 mm F/9 refractor of 2052 mm focal length. The effective focal ratio with the TZ4 is 9 x 4 = 36.0.

All refractors are stopped down to 50% aperture when used with the TZ2 in order to keep the same effective F/D ratio as with the TZ4.

**CWL shift (mica-spaced F-P etalon) :**

For the TZ2, the CWL shift from center to 16 mm from the optical axis is very small (lower than 0.1 A).

For the TZ4, the CWL shift is not noticable (lower 0.02 A).

**FWHM broadening with the TZ 2 (mica-spaced F-P etalon) :**

The 440 mm FL refractor is the one suffering the most from the use of a non-optimized telecentric system, with a broadening of the FWHM from center to 16 mm from center of about 0.15 A.

The FWHM broadening can be considered as negligeable or very small on the 1100 mm and 2050 mm FL refractors (resp. 0.05 A and 0.08 A)

**FWHM broadening with the TZ 4 (mica-spaced F-P etalon) :**

The greater magnification factor of the TZ4 (X4 instead of X2) results in a lower sensitivity to the focal length of the refractor. Any focal lengths over 440 mm will give excellent results.

**Conclusions on the use of non-optimized telecentric systems (mica-spaced F-P etalon) :**

Even if they are optimized for a specific focal length, telecentric systems can be used with some flexibility regarding focal length. Telecentric lens systems with higher magnification factor have more flexibility on the focal length.

Degradation of performance comes first for short focal lengths.

In order to benefit from the full performance of mica-spaced F-P etalons, the following requirements are to be met :

a) Use of a true telecentric lens system (such as Baader TZ or Beloptik). So called "telecentric Barlow lenses" are not telecentric and will give results of lower quality. The good news is that true telecentric lens systems give good performance, even when used at a focal length different to their designed focal length (within some limits ...)

b) Check out the the squareness of the F-P etalons to the optical axis. In other words, there should be no compromise on the mechanical set-up. As a rule of thumb the maximum allowance for the tilt is 0.1°.

It is a very easy to check out the squareness of the F-P etalon by auto-collimation :

- Place yourself at a few meters from the objective of your refractor.

- You should see : the cercle of the objective, the cercle of the etalon, and ... the reflection of your eye on the etalon.

- The reflection of your eye should be centered on the image of the etalon and on the objective.

Otherwise you should arrange the mechanical set-up.

c) The longer the focal ratio the better the performances are. F/D 30 seems to be a good starting point.