Sol'Ex Theory (SolEx)

A little instrument theory... Those who are less inclined to mathematics can skip this part, but that shouldn't stop them from building and using Sol'Ex! For everyone else, here's a rationale for the instrument's design, its size, and the origin of its power.

If Sol'EX is a spectroheliograph, it is first and foremost a spectrograph, i.e., an instrument that records the decomposition of light radiation into wavelengths, or colors, if you will. Christian chose the most classic and simplest optical arrangement available for Sol'Ex, as shown here.

The following figure shows the optical scheme:


 

The instrument consists of a narrow, 10-micrometer-wide, 4.5-mm-high entrance slit positioned at the focal point of the observation telescope. This is followed by an 80-mm-focal-length collimator lens consisting of two bonded lenses. This achromatic doublet was specifically optimized for Sol Ex and uses a special high-index glass. This lens causes the light rays emanating from a point on the slit to travel parallel to each other. The light rays then strike a holographic diffraction grating with 2400 lines/mm, which spectrally scatters the light. A 125-mm-focal-length lens, also specifically manufactured for Sol Ex, finally focuses all the rays in the detector plane.

The average direction of the rays before and after the grating forms a "V" with an interior angle of 34°. This angle is called the "total angle." It gives Sol Ex its characteristic overall shape.

It is immediately noticeable that the rays from the collimator hit the grating at a very steep angle, approximately 72° when working around the H-alpha hydrogen line at a wavelength of 656 nm. This strong incidence leads to strong vignetting of the rays collected by the telescope, with the grating, with its limiting size, roughly representing the pupil of the system. In the scattering plane (as shown in the figure), the aperture of the assumed beam is approximately f/10.6. In the perpendicular plane, and taking into account the area covered by the 4.5 mm high slit, the aperture of the system is approximately f/5.6. What does all this mean? Suppose you are using a telescope with a diameter of 65 mm and a focal length of 420 mm. The focal ratio of this telescope is then approximately 420 / 65 = 6.5. To the first order, when using the telescope with Sol'Ex, the effective aperture is f/6.5 along the axis perpendicular to the plane of incidence (the physical aperture of the telescope is the limit) and f/10.5 in the plane of incidence (Sol'Ex is the limit). If one could draw the actual contour of the usable area at the lens entrance, it would have the shape of an ellipse rather than a circle.

The dimensions of Sol'Ex therefore result in a loss of luminous flux because your telescope objective is stopped down (unless you are using a telescope that is natively opened to about f/10). This situation is not critical for solar observation, as the available luminous flux is truly abundant. Photographic optics and lenses that can be used with Sol'Ex can be opened between f/5.6 and f/9 without any visible effect on the images. However, if your instrument is very wide open, you can try different apertures in front of the lens to close it down and thus test for a possible improvement in image quality. I provide some examples in the "Observation" section. Conduct various experiments. This also prevents you from unnecessarily concentrating excessive light flux on a small point at the entrance of Sol'Ex. For example, if you want to use a photographic lens with an aperture of f/2.8, you should set the aperture to f/5.6 or even f/7.5 (diffraction generally does not degrade performance with regard to Sol'Ex's properties).

The strong incidence on the grating does cause flux loss, but it proves very beneficial in terms of the resolution of the acquired spectrum. To understand this last point, one must consider the significant difference in the size of the light beam in the plane of incidence in front of and behind the grating (D1 and D2, respectively):

This property, which is typical of the use of a grating, is called anamorphosis.

In addition to the size of the beams, anamorphosis also affects the size of the slit image on the detector. The image of the slit width is reduced by a factor of D1/D2, the so-called anamorphosis factor. However, the size remains unchanged along the so-called spatial axis, which runs perpendicular to the dispersion axis. To illustrate this, the following document shows an image of an optical fiber temporarily replacing the Sol Ex slit, with an initially round contour but an oval image at the end:

Abb.: Aussehen einiger monochromatischer Bilder einer am Eingang von Sol Ex angeordneten Glasfaser für Wellenlängen in der Nähe der H-alpha-Linie.

The reduction in the optical width of the slit caused by anamorphosis has a significant impact on Sol'Ex's spectral resolving power, i.e., on the fineness of the observable spectral details. In Sol'Ex's case, this impact is very positive. It is responsible for the high performance achieved despite Sol'Ex's compact size.

Let's do some calculations. First, a practical formula that relates the total angle G (in Sol'Ex G = 34°) and the angle of incidence alpha on the grating:

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

Where k is the diffraction order (here k = 1), m is the etching density (here m = 2400 lines/mm), and lambda0 is the wavelength at the center of the sensor (here 0.6563 x 10-3 mm). Furthermore, G = alpha + beta, where beta is the diffraction angle.

When observing the H-alpha line (6563 A), the angle of incidence of the rays on the grating is exactly alpha = 72.4°, while the diffraction angle (after the grating) is beta = 38.4°. The anamorphism factor is determined by the formula:

A = A = cos(alpha) / cos(beta).

which results in A = cos(72.4°) / cos(38.4°) = 0.386. This means that the width of the input slit is reduced (optically) to 0.386 x 10 micrometers = 3.86 micrometers. This result results in a gain in spectral resolution. It should be added that for the red hydrogen line, but also for a large part of the visible spectrum, the optics of Sol Ex are limited by diffraction, i.e., they are very good (unless one is simultaneously exploring a very large spectral range).

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

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

Where alpha and beta are the angles of incidence and diffraction on the grating, respectively. 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 in this formula, as is the effect of the grating's etching density. Performing the calculation around the H-alpha line, we find:

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

Taking into account the remaining optical aberrations, we can assume that the resolving power of Sol'Ex is close to R=40000, which is a remarkable achievement for such a small instrument. This means that, theoretically, spectral details of Δλ = λ / R = 6563 / 40000 = 0.16 A = 0.016 nm can be resolved in the red (but over a small wavelength range). Due to this fineness, the purity of the monochromatic images provided by Sol'Ex is better than that achieved with the much more expensive interference filters.

Let's recall the basic formula for gratings, which relates the diffraction angle to the angle of incidence:

sin(alpha) + sin(beta) = m x λ.

Where m is the etching density, here m = 2400 lines/mm, and λ is the wavelength, here λ = 0.6563e-3 mm. You can check whether the values ​​for alpha and beta are correct.

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

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

where p is the pixel size in millimeters and fo is the focal length of the camera lens, in this case fo = 125 mm.

Let's assume we are using a CMOS camera that uses the very attractive Sony IMX178 CMOS sensor (e.g., the ASI178MM camera at ZWO). The pixel size in this case is 2.4 micrometers. This gives us p = 2.4 micrometers = 0.0024 mm and,

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

Es ist wichtig, das zuvor berechnete spektrale Auflösungselement Δλ, 0,16 A, und die Abtastung des Spektrums durch die Pixel des Detektors, 0,063 A/Pixel, gegenüberzustellen. Es zeigt sich, dass man 0,16 / 0,063 = 2,53 Pixel pro Auflösungselement hat, d. h. man liegt über der Shannon- (oder Nyquist-) Grenze von zwei Abtastpunkten pro Auflösungselement. Dies ist eine gute Dimensionierung für Sol Ex. Beachten Sie, dass wenn die Kamera ASI178MM mit 2x2 Binning (entspricht 4,8 Mikron Pixel) betrieben wird, das Shannon-Kriterium nicht eingehalten wird und somit Informationen verloren gehen. Für Arbeiten, die eine hohe Genauigkeit erfordern (insbesondere Dopplermessungen), ist es besser, eine Kamera mit kleinen Pixeln zu verwenden und nach Möglichkeit mit 1x1-Binning zu arbeiten.

Diese Überlegungen zur Probenahme rechtfertigen voll und ganz die Verwendung eines Kameraobjektivs mit einer recht langen Brennweite von fo =125 mm. 

Eine weitere nützliche Formel ist diejenige, die die spektrale Dispersion in A / mm angibt, den sogenannten Plattenfaktor, der mit dem Buchstaben P bezeichnet wird:

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

Using the values ​​from our example, we find:

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

Now let's address the actual imaging of the Sun. As a first approximation, the sharpness of detail observed on the Sun's surface in arcseconds along the spectral axis is given by the width of the slit at the focal point of the telescope using the following formula:

Vx = 206264 x w / F.

Where w is the physical width of the slit and F is the focal length of the telescope. Assuming we are using a telescope 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 arcseconds.

In practice, the situation is more complicated, as pixel size and anamorphosis must also be taken into account, but the order of magnitude is good.

Depending on the spatial axis, the formula to use is:

Vy = 206264 x p x fc / fo / F

or

Vy = 206264 x 0.0024 x 80 / 125 / 420 = 0.75 arcseconds.

However, this result is very theoretical because it is geometric. In practice, given optical aberrations, one should expect a resolution of 3 arcseconds on the solar disk. It is also quite common to work with 2x2 binning, and in this case, the effective pixel size is p = 2.4 = 4.8 micrometers.

Ultimately, one can therefore assume that the spectral resolution in this example is approximately 3 arcseconds. Atmospheric turbulence can certainly reduce this performance. On a hot day, seeing can easily exceed 3 arcseconds. If your seeing is better, you can increase the angular sharpness of the images by increasing the focal length of the observing telescope (I emphasize this in the "Observing" section).

Finally, you need to answer the question: Can I capture the image of the entire solar disk in a single sweep using a telescope with a focal length of F?

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

Limiting F_F = 4.5 / tan(0.53°) = 486 mm.

However, it is advisable to allow for a margin of at least 10% to properly frame the disk over the length of the slit. Therefore, an F-stop focal length of 440 to 450 mm is a reasonable maximum. Furthermore, the edges of the slit are slightly less sharp than the center, so blurring may occur at the poles of the disc when panning straight up. Ultimately, a focal length close to 420 mm is certainly ideal if you want to comfortably capture the entire disc. Of course, the focal length can be much longer if this isn't your priority because you want to capture the solar surface in as much detail as possible.

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Graphics and images were adapted from Christian Buil, who developed the Sol'Ex.