## Collimating with a collimator27 January 2020 (© N. de Hilster & Starry-Night.nl) Adjusting the collimation of mirror-type telescopes like Schmidt-Cassegrain (SCT) and Ritchey–Chrétien (RC) is a regular recurring job. All that is needed for this, apart from the scope itself, is a point source. This can be an ordinary star, an artificial star or a collimator. To collimate using a real star, however, clear weather is needed, preferably with perfect seeing, exactly the kind of conditions where you want to observe or photograph rather than collimate. The use of an artificial star is a good alternative, but requires a minimum distance that is not always easy to achieve indoors. A collimator has the advantage over the two previous methods that collimation can be done indoors at any time and that the collimator does not have to be further away from the scope to be collimated than is necessary to reach its adjustment screws (i.e a few decimetres). This article is about the basics of collimation, the construction of a collimator based on a Newtonian mirror and its use. ## The theory behind the collimatorAn 11″ Schmidt-Cassegrain is in use on InFINNity Deck (Celestron C11 EdgeHD). During cloudy periods it could be collimated by using an artificial star. However, it must be at a minimum distance that meets (Suiter 91): Distance_{Art.star - Scope} = 336 x D / F^{3} x focal lengthwhere D is the diameter of the scope in inches and F its focal ratio. Since this is an absolute minimum, it is recommended to double or triple the result (Suiter 91-92). However, it is also advised to use a factor of at least 10 as an outcome and then to double or triple it, so in fact a minimum factor of 20 (Suiter 91-92). So for an 11″ f/10 SCT, the distance becomes 336 x 11 / 1000 = 3.7 x focal length. According to advice, this should be doubled (so 7.4 x) and the factor should not be less than 20 (so it will be 20 x). To be able to test the 11″ SCT with an artificial star, 20 x focal length, or 20 x 2.8 = 56 meters, must therefore be maintained, something that is not always practical. That is why I decided to build my own collimator, since the distance will no longer play a role. According to the Oosthoek Encyclopedia from 1916, a collimator is an “Optical device for obtaining a parallel beam of light”. The light that reaches our telescopes from the stars is parallel on the scale of our telescopes, just like a collimator. And that is precisely why the distance between collimator and telescope no longer matters. A very simple collimator can be seen in figure 1. This consists of a light source that is placed exactly in the focal point of a biconvex lens. The light passing through this lens will come out parallel on the other side. In order to collimate a telescope using a collimator, the collimator should preferably have at least the same diameter as the telescope so that the entire mirror of it participates in the image forming. Preferably, the collimator should be slightly larger, so that the correct positioning of the telescope to be collimated is not critical. So the C11 from InFINNity Deck would require a lens of at least 11″ (279.4mm). Apart from the fact that such a lens is not easy to obtain, an ordinary biconvex lens also has the disadvantage that spherical and chromatic aberration will occur. Of course a monochromatic light source can be chosen to prevent chromatic aberration, but the spherical aberration is less easy to solve. A good alternative to the biconvex lens is a mirror (see figure 2). Mirrors of the diameter mentioned above are widely available, both as a separate component for self-build and in the form of a Newton telescope. Chromatic aberration will not occur, but spherical aberration (ΔW _{sph}) will if a too fast spherical mirror is chosen. The calculation on which this is based concerns a series development (Wyant 50-51):In this formula, s is the radius of the mirror (s in figure 3) and f# is its focal ratio (f# = focal length / mirror diameter). The result is the difference between the spherical mirror and parabolic mirror in micrometers (µm). Assuming green light with a wavelength of 0.55μm (550nm), the aberration can then be found by dividing it into units of wavelengths (λ): ΔW_{sph} (λ) = ΔW_{sph} (μm)/0.55Now this calculation is not easy to explain in a picture. The calculation is clearer using the sag (z in figure 3). The sag is the difference in height between the centre of the mirror and its edge (the diameter of the mirror is D in the figure). The spherical aberration is now found by subtracting the sag of the parabolic mirror from that of a spherical mirror. Since the light must cross this space twice to get from the focal point to the telescope to be collimated, the difference found must be multiplied by 2. Division by the wavelength of the light gives the result in wavelengths. The sag z of the parabola and the sphere can be calculated as follows (Smith et al. 59-61, see figure 4): Here c is the reciprocal of the radius of curvature in the vertex (center) of the mirror: c = 1/R (R = 2 x focal length), h the radius of the mirror (s in figure 3), and k is the conic constant (k=0 for a sphere, k=-1 for a parabola, see figure 3). The result is in the same units as R and h are given. From these two sag formulas, the spherical aberration follows as (assuming both sag are calculated in µm): _{sph} (λ) = 2 x (z_{s} – z_{p}) / 0.55For a large fast mirror (12″ f/4) the spherical aberration is 16.945λ, with the difference between the two methods occurring from the 5th decimal place and thus negligibly small. At my request, the calculations for this theoretical mirror have been verified with optical design software ZEMAX 13. The value found by Ray tracing is 16.879λ (see figure 5), a still negligible difference of 0.066λ. Figure 6 shows the spherical aberration as a function of focal ratio and mirror diameter. The diagonal lines represent the different spherical mirrors, while the horizontal red line represents a quarter lambda criterion within which the spherical aberration should remain for the collimator to function properly. Figure 6: Paraxial-focus spherical aberration of spherical mirrors at different diameters and focal ratios.It is clear that the spherical mirror must be at least f/11 to be usable as a collimator, but then it must remain well smaller than 4″. To build a good collimator for a C11, an 11″ f/16 spherical mirror would be needed (not shown in the graph). However, with a length of almost 5m, this becomes impractical (not to mention availability) and so the use of a parabolic mirror is preferred. The above calculations have been made for the so-called paraxial focus point, the point at which the centre of the spherical mirror comes into focus. This automatically follows from the above geometric sag-difference calculation. ZEMAX and Wyant both use this focal point in their calculations (ZEMAX has several other focal point settings besides this, such as smallest PV spot size, smallest rms spot size , and smallest rms wavefront error). FigureXP and DFTFringe, two software packages for testing mirrors, use the best-focus point, which is closer to the mirror than the paraxial-focus point (Sacek 4). Figure 7: Best-focus spherical aberration of spherical mirrors at different diameters and focal ratios.The best-focus point is the point at which the diffraction disk is smallest (this corresponds to the part of the mirror at 0.707 x the mirror radius), in practice we speak of “the telescope is in focus” (Suiter 187). The difference in spherical aberration (ŵ) between these points can be calculated from ŵ=√(1+0.9375Λ(Λ-2)) with Λ=0, 1, 2 for paraxial-focus, best-focus and marginal - focus respectively (the point where the edge of the mirror comes into focus, Sacek 4). The best focus point is the middle point between the marginal and paraxial focus (Sacek 4). As best focus means Λ=1 and thus ŵ=0.25, so exactly a factor 4 smaller than paraxial-focus. Marginal focus is at Λ=2, which makes ŵ=1 and so the aberration there is equal to that of the paraxial-focus (Sacek 4). For the graph in figure 6, this means that all lines become flatter and when using best-focus, the spherical mirror should be at least f/7 (maximum 4″ diameter, see figure 7). For a 300mm (12″) collimator with a spherical mirror, at least an f/11 mirror would be required to keep the spherical aberration within a quarter lambda. Whichever method is used, a spherical mirror is not very practical for a collimator. ## The construction of the collimatorInspired by a similar solution I had seen at the Almere public observatory, combined with the above reasons and advice from Jan van Gastel (working at the same observatory), and due to chance availability, I finally opted for a SkyWatcher Explorer 300PDS Newton (see figure 8). This telescope has a parabolic mirror of 304mm diameter and a focal length of 1500mm (f/4.9) and was recently for sale via a forum. Next, a small light source is needed that will act as an artificial star in the focus of the telescope. Inquiries with Jan van Gastel learned that they use a 9μm fiber optic cable, coupled to a bright red LED. He didn't have any further specifications, so I asked my astro-friend Rob Musquetier if he could provide me with a schematic for dimming LEDs in case the ordered LEDs were too bright. The artificial star now consists of a 12V circuit (see figure 9) that can switch on/off and dim three different LEDs via a switch. I chose three LEDS as I wasn't sure what would work best so now I can experiment. The first LED is a matte red, the second a bright red, the third a switchable RGB LED. Figure 10: The circuit in the housing with fiber optic cable and 1.25″ socket in front of the focuser.The LEDs are 5mm in diameter and are mounted in ST connectors, on which a Duplex fiber optic cable ST/ST 9/125 1m OS2 can be plugged. This cable has a core diameter of 9μm and is fitted with another ST coupler in a socket I created on my lathe and fits the focuser of the SkyWatcher 300PDS Newton (see figure 10). ## Using the collimatorNow that the parts are ready, the collimator can be used. What is needed for this is a telescope that can be collimated, a Barlow (a 2x one will suffice), an eyepiece with as long a focal length as possible, and a camera with connection to the focuser of the telescope to be collimated. The idea is quite simple: the collimator and telescope are placed opposite each other and properly aligned (see figure 11). If this is the case, the artificial star will be visible with the telescope to be collimated. The long-focal eyepiece is useful here, as it produces a large field of view and makes it easier to find the artificial star. A pair of wedges under the telescope help to tilt the telescope in the right direction, while for lateral corrections the telescope is simply shifted (rotated in the horizontal plane). Subsequently, with a few final minor corrections, the artificial star is centred as well as possible in the eyepiece, and the camera with Barlow takes the place of the eyepiece. The use of a camera is not strictly necessary, but given the distance between the adjustment knobs on the front and the eyepiece on the back of the telescope, it is easier to collimate the telescope while viewing the image from the camera on a screen. Thanks to the adjustable brightness of the LEDs, the exposure time of the camera does not need to be adjusted when focusing in and out. Incidentally, the artificial star can also be found without an eyepiece by simply looking through the hole of the focuser. Figure 12: Artificial star in focus with diffraction pattern. The spikes are from the spider of the collimator.At the beginning of this article I wrote that the artificial star should be sufficiently far away. However, compared to the stars, this minimum distance is relatively close. If the telescope is roughly focused on the starry sky, then that is sufficient. Optionally, the telescope can also be focused on surrounding objects (e.g. distant trees or buildings) before placing it opposite the collimator. If the telescope is already well focused at infinity, the focuser of the collimator can be adjusted until the artificial star has minimal dimensions and the diffraction pattern becomes visible (see figure 12). The focuser of the collimator can then be marked so that it will immediately be at (approximately) infinity the next time. The collimation of the collimator itself also turned out not to be very critical. It has been established experimentally that even a deviation of a few centimetres from the laser line of a laser collimator, mounted in the eyepiece of the collimator, does not appreciably affect the image of the telescope to be collimated. To now adjust the telescope to be collimated, it must be rotated 8 to 12 wavelengths out of focus (Suiter 88). How much this is in millimetres can be calculated with the following formula (Suiter 84): ^{2}ΔnλWith F the focal ratio of the telescope (including a possible Barlow), λ the wavelength of the light in μm and Δn the number of wavelengths one wants to turn the telescope out of focus. An f/10 telescope with 2 x Barlow (the combination is then f/20) with green light (0.550μm) can therefore best be assessed with a defocus of 14 to 21 millimetres. In the case of a C11 EdgeHD with MoonLite focuser, this means that the entire range of the focuser may be used. The artificial star will then look like a doughnut. If the telescope is out of collimation, the central hole of the doughnut will not be in the centre of the circumference (see figure 13). Correcting an SCT is done by turning the knobs on the secondary mirror so that it changes position. Not only will the doughnut look different as a result, but its position on the imaging chip will also shift. So any correction requires turning the telescope to centre the image again. Figure 14: The direction the doughnut moves when adjusted is the direction the doughnut hole will move.Based on the movement of the doughnut on the screen, it can be immediately deduced whether the correction is being made in the correct direction (see figure 14). If the hole has to move to the top left, the entire doughnut will have to move that way. Which screw needs to be tightened or loosened for this can be determined by pointing it out by hand in between the (artificial) star and telescope. The hand will cast a shadow in the doughnut and serve as a reference. Once properly collimated, the telescope will show the artificial star as a perfectly symmetrical doughnut (see figure 15). This is a good time to rotate the telescope 180° along the optical axis to see if mirror-flop (tilting one of the mirrors due to structural play) affects collimation. If this is the case, then half of the error can be tuned away to have a reasonably collimated telescope on average. It is of course better, if possible, to eliminate the cause of the mirror flop. To make collimating easier I wrote a piece of software, CollimatorGrabber, that cuts the image in parts, mirrors them and recombines them to show the differences between them. Finally, an animation shows the advantage of the collimator over a real star (see figure 16, the green cross comes from CollimatorGrabber and shows the axes along which the image is cut and mirrored). Although air vortices can also have a disruptive role in a collimator, their influence is not nearly as great as in a real star. Figure 16 shows the doughnut of a C11 EdgeHD using a collimator (left) and a real star, in this case Capella at about 60° altitude and a seeing of about 2″. The collimator produced first-light on December 30, 2019 with first telescopes Rob Musquetier's RC10 and AWSV Metius' C11 XLT. Later it was used to collimate an RC8 and a further two RC10s. This article was made possible in part thanks to the input of Jan van Gastel (Almere Observatory) and Roger Ceragioli (Richard F. Caris Mirror Lab, University of Arizona, Tucson (USA)). ## SourcesV. Sacek, Notes on AMATEUR TELESCOPE OPTICS, (Internet, 2006), https://www.telescope-optics.net/, last accessed July 20, 2023. G.H. Smith, R. Ceragioli, R. Berry, Telescopes, Eyepieces, Astrographs: Design, Analysis and Performance of Modern Astronomical Optics, (Richmond (VA), 2012). H.R. Suiter, Star Testing Astronomical Telescopes: A manual for optical evaluation and adjustment, (Richmond (VA), 2013). J.C. Wyant, “Basic Wavefront Aberration Theory for Optical Metrology“, in: R.R. Shannon, J.C. Wyant, Applied Optics and Optical Engineering, vol. XI, (1992), pp.50-51, last accessed July 20, 2023. If you have any questions and/or remarks please let me know. |

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