Theory

The reduction in the optical width of the slit caused by the anamorphosis has a significant net impact on the spectral resolving power of Sol'Ex, that is to say on the sharpness of the details of the spectrum that can be observed. In the case of Sol’Ex, this impact is very positive. He is responsible for the high performance achieved while Sol’Ex is a compact instrument.

Let's do some math. First the link between the total angle G (G = 34° for Sol’Ex)  and the incidence angle alpha on the grating  :

alpha = arcsin(k m lambda0 / (2 cos G/2) + G/2

where k, is the diffraction ordre (hère k = 1), m, the groove densité (ici m = 2400 lines/mm) and lambda0, the wavelength at the sensor center (here 0,6563x10-3 mm). We have also, G = alpha - beta, with beta, the diffraction angle.

For the observation of the H-alpha line (6563 A), the incidence angle of the rays on the grating is precisely alpha = 72.4° while the diffraction angle  (after the grating) is beta = 38 , 4°. The anamorphosis factor is given by the formula:

A = cos (alpha) / cos (beta)

where A = cos (72.4 °) / cos (38.4 °) = 0.386. This means that the width of the entrance slit is reduced somewhat. (optically) at 0.386 x 10 microns = 3.86 microns. It is this result that causes a gain in spectral resolution. It should be added that towards the red line of hydrogen, but also a large part of the visible spectrum, the optics of Sol'Ex is almost limited by diffraction, that is to say it is very good ( since one does not simultaneously explore a very wide spectral domain).

The resolving power R is defined by the formula R = λ / Δλ, with λ the observation wavelength and Δλ the finest detail observed in the spectrum in units of wavelength. Note that R is a dimensionless number. The larger is R, the more small details are observed in the spectrum. We show that:

R = fc / w x (tan (alpha) + sin (beta) / cos (alpha))

as we have seen, alpha and beta, respectively the angle of incidence and diffraction on the grating. Furthermore, fc is the focal length of the collimator, here fc = 80 mm, and w is the physical width of the slit, here w = 10 microns = 0.010 mm. The phenomenon of anamorphosis is described by the term in parentheses of this formula, as well as the impact of the etching density of the grating. By doing the calculation around the H-alpha line, we find:

R = 80 / 0.010 x (tan (72.4 °) + sin (38.4 °) / cos (72.4)) = 41600

Considering the residual optical aberrations, we can consider Sol’Ex's resolving power to be close to R = 40,000, which is a remarkable performance for such a small instrument. We therefore theoretically solve in the red (but over a small width in wavelength) spectral details of Δλ = λ / R = 6563 / 40,000 = 0.16 A = 0.016 nm. This fineness makes the purity of the monochromatic images delivered by Sol'Ex higher than that obtained with interference filters, much more expensive.

Recall the fundamental formula of gratings, which relates the diffraction angle of diffraction to the incidence angle:

sin (alpha) + sin (beta) = m x λ

With m, the engraving density, here m = 2400 lines/mm and λ the wavelength, here λ = 0.6563e-3 mm. You can check that the alpha and beta values are correct.

Another important optical parameter is the spectral dispersion factor in the plane of the detector. It is a question of evaluating the small spectral range covered by a pixel of the detector. This parameter, denoted by r, if it is expressed in A/pixel (strictly speaking, it is a reciprocal dispersion), can be calculated with the formula:

r = 1e7 x p x cos (beta) / m / fo

with p the pixel size in millimeters and fo the focal length of the camera lens, here fo = 125 mm.

Suppose a CMOS camera using a Sony IMX178 CMOS sensor (ASI178MM camera from ZWO for example). The pixel size in this case is 2.4 microns, where p = 2.4 microns = 0.0024 mm and,

r = 1e7 x 0.0024 x cos (38.4 °) / 2400/125 = 0.063 A / pixel

It is important to compare the spectral resolution element Δλ calculated previously, that is 0.16 A and the sampling of the spectrum by the pixels of the detector, 0.063 A/pixel. It appears that we have 0.16 / 0.063 = 2.53 pixels per element of resolution, that is to say that we are above the Shannon (or Nyquist) limit which is two sample points per element of resolution. This is a good sizing for Sol’Ex. Note that if the ASI178MM camera is operated in 2x2 binning (equivalent to 4.8 micron pixels) the Shannon criterion is not respected, and therefore information is lost. For jobs that require high precision (Doppler measurements in particular), it is best to use a small pixel camera and work in 1x1 binning, if possible.

These sampling considerations fully justify the use of a fairly long focal length camera lens of fo = 125mm.

Another useful formula is the one that gives the spectral dispersion in A / mm, called the plate factor, denoted by the letter P:

P = 1e7 x cos (beta) / m / fo

With the values of our example we find,

P = 1e7 x cos (38.4) / 2400/125 = 26.1 A / mm

Now let's deal with the subject of Sun imagery itself. As a first approximation, along the spectral axis, the sharpness of the details observed on the surface of the Sun in arc second is given by the width of the slit at the focal point of the telescope using the formula:

Vx = 206264 x w / F

With w the physical width of the slit and F the focal length of the telescope. Suppose we are using a refractor with a focal length of 420 mm, in this case with w = 0.010 mm, we find:

Vx = 206264 x 0.010 mm / 420 = 4.9 arc seconds

In fact, the situation is more complicated, because you also have to take into account the size of the pixels and the anamorphosis, but the order of magnitude is good.

Along the spatial axis, the formula to use is,

Vy = 206264 x p x fc / fo / F

that is

Vy = 206264 x 0.0024 x 80/125/420 = 0.75 arc seconds

However, this result is very theoretical because it is geometric. In practice, taking into account optical aberrations, it is more necessary to count on a resolution of 3 seconds of arc on the solar disk. Likewise, it is quite common to work in 2x2 binning and in this case, the effective pixel size is p = 2.4 = 4.8 microns.

In the end, it is reasonable to assume that the spectral resolution is close to 3 arc seconds in this example. Atmospheric turbulence may well reduce this performance. On a hot day, seeing can easily exceed 3 arc seconds. If your seeing is better, the way to increase the angular resolution of the images is to increase the focal length of the refractor (I emphasize this point in the "Observation" section).

Finally, the question must be answered: can I capture the image of the entire sun disk in a single scan using a telescope with focal length F?

The apparent diameter of the Sun changes slightly depending on the season, but we consider here the diameter of 0.53 °, very representative. Since the length of the Sol’Ex slit is 4.5 mm, the desired limit focal length is given by the formula:

F_limite = 4.5 / tan (0.53 °) = 486 mm

However, it is prudent to take a margin of at least 10% to reasonably frame the disc over the length of the slit. So, focal length F of 440 to 450 mm is the reasonable maximum. Also, the edges of the slit are a little less sharp than the center, resulting in a potential little blurring at the poles of the disc if you are sweeping in right ascension. Ultimately, a focal length of around 420mm or less is ideal if you want to capture the entire disc with comfort. Of course the focal length can be much longer if this last point is not your priority, because you want to detail the solar surface as much as possible.