Backyard Voyager
The Autocollimator and its reflections Nils Olof Carlin
The Autocollimator and its reflections
I have long had a feeling that the existing descriptions of the autocollimator and its mode of operation have not been based on a clear understanding. This analysis attempts to describe the reflections seen and their positions as functions of miscollimation and builds on a discussion in the Yahoo Collimate_Your_Telescope forum.
The autocollimator is a flat mirror mounted in a short tube made to fit a Newtonian telescope focuser, and set accurately perpendicular to the tube’s axis. Centered in it is a small peephole or pupil that you look through. If the primary mirror’s center is marked with a bright or reflective spot, you can see the spot (reflected in the secondary) and a few more reflections of the spot after several reflections back and forth. This picture shows a 2” autocollimator (INFINITY, ™ by Jim Fly).
To use it, you first do a fairly close collimation with a sight tube and a Cheshire collimator (or, if you prefer, with a laser with a Barlow attachment), then insert the autocollimator and fine-adjust the collimation. When collimation is ideal, the reflections will appear “stacked” or coincident – but if they are not, how do you proceed? To answer this vital question, I believe you need a clear understanding of how the reflections are generated.
This figure (click this or later images for a larger version!) shows, greatly exaggerated and not to scale, the primary and autocollimator (the secondary mirror is ignored, not to introduce even more confusion). The primary’s optical axis starts at the center mark P and goes to COC, the center of curvature of the mirror. Midway between P and COC is F, the focus of the primary mirror.
The autocollimator’s axis is perpendicular to the mirror and centered in it, at the pupil. There is a combined collimation error shown: the primary’s axis misses the autocollimator pupil by a distance A (elsewhere I have called this a 1A error!), and the autocollimator(=focuser) axis misses the primary’s center by B (1B error).
To trace the reflections:
Draw a line (green) from P, parallel to the autocollimator axis, to V1 via H5, and another parallel line from F to H3. Also, draw a line parallel to them from COC to H2, and a third line V2 to “2”, parallel and with a distance V2-COC equal to COC-V1.
At V1 is a virtual image of P by the autocollimator. This virtual image is reflected to a focused virtual image (the details are not shown) V2, also at the level of the COC. This virtual image is again reflected in the autocollimator down to a real image “2” at the primary. This reflection can be seen as an “inverted”, or more accurately rotated by 180°, but (just like P) sharp when seen with an eye or camera focused at one focal length.
Also another reflection “1” can be seen (this will be shown later), by intercepting the light going towards the image V2 – its projected position on the primary is found by drawing a line V2 to autocollimator pupil, extending it down to the primary.
The real image at “2” is projected by the reverse path back to P, accurately on top of it after another 180° rotation, thus ending for good the series of reflections. But before this, a reflection “3” can be seen, formed from “3” in analogy to how the reflection “1” is formed from P. Both “1” and “3” are seen as bundles of converging light, as if focused at minus one focal length – if imaged with a camera focused at P, they will appear noticeably defocused (if the camera is focused at infinity, all images will appear defocused by the same amount).
Now we can determine the relative positions of the reflections (positive to the right in the figure):
The distance P to H3 is A+B, as is H3 to H2. H2 to “2” equals p to H2 =2A+2B, the total displacement P to “2” is 4A+4B.
By similar reasoning, the distance from H4 to the autocollimator pupil is 4A+3B, and from “2” to “1” twice this – thus the displacement of “1” from P is -4A-2B. The displacement of “3” from P is 2B, remarkably enough independently of any miscollimation of the primary.
This figure shows the paths of reflections “1” (bold red) and “2” (bold green) seen in the autocollimator after 2 and 4 reflections respectively (note that the reflection at H6 should rightly have occurred much farther to the right!). The reflections of “3”, (if you include the reflections in the secondary, there are 13!) are left as an exercise to the reader.
Let us also regard the reflections of the autocollimator:
Draw a line from COC through the ACP, it reaches the primary at X, displaced -2A from P. This is where the autocollimator will be seen by its first or "foreground" reflection – in the same manner as the face of a Cheshire!
Draw a line (green) from F parallel to the autocollimator axis down to G1.
Trace a ray from ACP parallel to the primary’s axis down to G2 – it will be reflected toward F, and then to G3 where the distance G3-G1 = G1-G2=2A+B, and then parallel to the focuser axis up to G4 where the distance G4-F=G3-P=3A+2B. Thus, there will be a real "background" image of the autocollimator pupil at G4, displaced from the "foreground" reflection by 4A+2B (and the whole autocollimator mirror around it, rotated 180°).
The reflection at G4 will be imaged back to the ACP itself, at least as long as G4 still falls on the mirror face! Thus, the light path will be closed and the autocollimator face will be dark regardless of collimation – “darkening” is not a useful collimation criterion as has sometimes been claimed.
The reflections of the pupils will be visible against the background of more or less coincident spot reflections - but not easily so, if it were indeed possible to stack the reflections perfectly.
The "background" reflected pupil at G4 will appear at a displacement of 2A+2B from P (not shown).
Miscollimation errors:
Given a miscollimation (in one dimension of two!) of A of the primary axis and B of the focuser/eyepiece axis, the displacements of the reflections are:
| P to “1” | -4A-2B |
| P to “2” | 4A+4B |
| P to “3” | 2B |
| P to X (the “foreground” reflection) | -2A |
| P to the “background” reflection | 2A+2B |
| “foreground” to “background” reflections | 4A+2B |
You can also use the angular error expressed as C=A+B:
| P to “1” | -2C-2A |
| P to “2” | 4C |
| P to “3” | 2C-2A |
| P to X (the “foreground” reflection) | -2A |
| P to the “background” reflection | 2C |
| “foreground” to “background” reflections | 2C+2A |
Here, you see that while the sensitivity to angular errors is high (P to “2” = “3” to “1” = 4C), the sensitivity to errors of the primary’s axis is not greater for the autocollimator than it is for the Cheshire or Barlowed laser.
What it looks like in the autocollimator:
These images by Vic Menard show eloquently what can be seen: The camera is focused at P, so it and the rotated image “2” are sharp, while “1” (non-rotated, bright) and “3” (rotated, fainter) are defocused.
In the image above, the focuser (or secondary) is intentionally misaligned (A=0), so you see from left to right:
Also note the reflection of ACP at +2B, falling on “3” making the center hole appear sharper than the edge. |
In the image above, it is instead the primary that is miscollimated (B=0), so you will see from upper left to lower right:
(There is also a small miscollimation at right angles to the main one) |
In principle, it is possible to solve an equation system for each direction: call X=P to “1”, Y=P to “2”:
- A=-X/2-Y/4;
- B=X/2+Y/2;
But this doesn’t seem very practical out in the field! And the appearance of the spots doesn’t in general suggest what needs adjusting – trying to stack P and “2” will leave you with A=-B and “1” and “3” slightly off by 2B in opposite directions.
One theoretical solution: offset the primary collimation enough to separate the spots, then identify the faintest reflection “3” and stack it with P by adjusting the secondary (thus setting B=0), and finish by adjusting the primary until all spots stack. Vic Menard reports that it is indeed a useful method! But you might want to finish by checking with the Cheshire, anyway.
However, I believe (along with Vic Menard and Jim Fly) that the most practical approach is to use the sight tube (or laser) to collimate (at least roughly) the focuser axis (by adjusting the secondary mirror), then use the Cheshire (or Barlowed laser) to adjust the primary as accurately as possible. Thus, with A=0 set as closely as possible, you adjust the focuser tilt (if adjustable!) or the secondary to stack all images, finishing by checking that A=0 still – if not, repeat. This will converge quickly, quicker if you can adjust the focuser tilt rather than the secondary tilt.
What precision could be expected in practice?
The fundamental weakness - often touted as the main advantage - of the autocollimator is its "multi-pass" nature: the errors of the axes are intermixed (but for the faintest reflexion "3", only separately discernible by intentional miscollimation of the primary) and cannot be separated for separate adjustments. Assume you have an error ΔA after collimating with the Cheshire, and succeed in bringing the reflections “1” and “2” accurately together (they will be removed from P by a minimum distance of ≈ ΔA): thus -4ΔA-2ΔB=4ΔA+4ΔB or ΔB=-4/3×ΔA). The residual error after bringing P and “2” together is of the same magnitude.
What about reading accuracy? If you use a Cheshire with a fine peephole, you see the spot (at one focal length) and reflected bright face of the Cheshire (at infinity) sharply, independently of the eye position, but if you enlarge the peephole, you will find the lineup shifting slightly if you move your eye (in the Barlowed laser, the very small point source of light corresponds to a very small peephole of a Cheshire). With an autocollimator with a fairly large peephole (see the first illustration above), you will see the reflections P and “2” at one focal length, their separation independent of the eye position, but the reflections “1” and “3” are defocused (twice as much as in the case of the Cheshire) and may be affected by the eye position with some loss of reading accuracy. However, the distance between the “foreground” and “background” reflections of the autocollimator is equal to the distance between P and the defocused “1”, but here they are both focused at infinity and free of parallax - thus, an autocollimator with the inner edge of the barrel coated with reflective material may show this separation to higher precision by eliminating the "lune" from the "background" reflection of the autocollimator edge.
If the "foreground" reflection of the autocollimator pupil itself can be seen against the background of slightly displaced and defocused spot reflections (and I expect this is unavoidable in practice - but there is also a "background" reflection of it that may interfere), it can of course be read in the same manner as a Cheshire (with some reservation for the size of the peephole, see above), eliminating the swapping of tools at least until the very final stages.
Given one of the axes (usually the primary's) is collimated to within a small residual error with an independent method, the autocollimator can be expected to bring down the collimating errors of the other axis to the same order of magnitude, but not better - even considering the limits to its own reading precision.
One trap to avoid: adjusting the secondary will affect the collimation of the primary's axis to some extent, and if you correct a large error B this way, you may introduce a significant error A - thus, do not forget to check with the autocollimator/Barlowed laser after any (unless minor) adjustment of the secondary.
My final words:
The autocollimator may be quite useful for finding and correcting potentially significant errors of the focuser axis, left over from "eyeballing" or using a sight tube of limited precision. Whether the residual errors from collimating the focuser axis with a laser collimator are large enough to affect the image may depend on the particular application - if nothing else, a discrepancy would indicate the need to check the collimation of the laser or autocollimator itself (even a tiny speck of dust between the edges of the focuser and autocollimator or laser may have quite obvious effects!). However, you should not expect an improvement of the generally more critical collimation of the primary over what could be achieved with a well functioning Cheshire or Barlowed laser.