The Autocollimator (AC) and its reflections

With new comments, July 2010

The original version of this webpage was, I believe, the first reasonably complete analysis of the operation of the autocollimator. Since then, the understanding of the autocollimator, its operation and its use - and also its design! - has been further enhanced by the work of Jason Khadder, "Jason D" on the Cloudy Nights' Reflectors forum:

This analysis attempts to describe the reflections seen and their positions as functions of miscollimation and builds on a discussion in the autumn of 2004, 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. (The mirror should be placed reasonably close to the actual focal plane of the telescope. In practice, if you position the focuser in the range you use for observing with any of your eyepieces, this is close enough). Centered in it is a small peephole or pupil that you look through. The primary mirror’s center is marked with a bright or reflective spot, and 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 the model XLK autocollimator, with the ordinary, centered pupil (to the left) and the added, decentered pupil (INFINITY, ™ by Jim Fly). The mirror is highly reflective, and thus almost impossible to see, but it is placed midway between what appears to be the open ends of the aluminium barrel!

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 or without a Barlow attachment), otherwise the reflections may fall well outside where you can see them.

Then insert the autocollimator and fine-adjust the collimation. Change between the AC and the Cheshire as in the example below, until the reflections show that you are close enough. But what exactly are the reflections, and what do they show?

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 secondary, there are 13 reflections in all!) 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 primary's 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:
  • “1” at -2B
  • P
  • “3” at 2B
  • “2” at 4B

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, so you will see from upper left to lower right:

  • “1” at -4A
  • P and “3” almost coincident
  • ,
  • “2” at +4A

There is also a smaller miscollimation of B at almost right angles to the main one - when the reflections are well separated as here, it is easy to see that "3" is slightly below and to the left of "P". What you can do here is adjust the secondary to bring "3" to coincide with "P" (never mind what happens to "1" and "2"!) This done, you know that any error left is in the collimation of the primary. Adjust it until the reflections ("1" and "2") merge with "P" and vanish - now collimation is done! This is, in fact, how the "carefully decollimated protocol" works. To use this method, you may have to first slightly decollimate the primary in order to separate the reflections and identify "3".

In principle, it is possible to solve an equation system for each direction: call X=P to “1”, Y=P to “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.

During the 4 years from the first version of this webpage, this method (sometimes known as "the Carefully Decollimated Protocol" has gained popularity as a quick and useful method.

One 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. (The CAM, briefly mentioned above, makes the AC work like a Cheshire, thereby eliminating the need for swapping).

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.

The Case of the Missing Reflections:

When you come close to convergence, the "inverted" reflections (2 and 3) seem to vanish - you would expect all reflections to stack neatly and add up to a regular hexagram star pattern, but instead you only see one triangle (P, actually). Why?

The simple answer: at convergence, the reflections have to pass the AC twice along quite symmetric paths. The first time outside the pupil in order to be reflected further, the second time inside the pupil in order to be observed. This is impossible of course, and the reflections will vanish. However, this problem can be solved by adding a second, decentered pupil: looking through it instead, you see the reflections P and 2 stacked, and separately 1 and 3 stacked.

For a complex answer, look at the figure: we will trace a sample ray from the tip of the triangle to its expected (but missing) reflection 2 and further to the observer's eye. Call R= the radius of the AC pupil.

The tip is at -S, and we will use an arbitrary parameter Y, such that the first reflection (1) in the AC occurs at -S+Y.The next reflection (2) is at the primary, at (-S+2Y).

Next reflection again (3) is at the AC, at (S-Y) - and the next in turn (4) is at the primary at (you may have guessed) S. But from here, it goes back to the AC (5) and hopefully out the pupil, to let you see it.

Now, there are several hurdles to pass: the first reflection must not hit the AC pupil, in order to go on:

(-S+Y)<-R i.e. Y<(S-R) if to the left of it in the figure, or

(-S+Y)>R i.e. Y>(S+R) if to the right.

The next hurdle is of course hitting the triangle itself, but we'll leave this for a while...

Then the ray must hit the AC pupil in order for us to see it: -R<(S-Y)< R , i.e.

Y<(S+R) and Y>(S-R)

- in other words, when collimation is done, no light at all can go from the triangle to form the real reflection 2 and still reach your eye. This is independent of the size of the triangle, as well as of the autocollimator pupil.

However, the necessary and sufficient condition for the reflections to vanish is that the AC axis intersects the COC - this means that P and 1 are aligned, and thus 2A=-B, but it does not mean that both are zero. Thus, you can not take the vanishing of the reflections as a sign of true collimation - you need to check with the Cheshire (or equivalent) to verify that A=0.

The disappearance is gradual when the above condition is approached. Also, the above assumes a circular pupil and the AC mirror at the focal plane.

This series of images by Jason Khadder (used here by permission) illustrates some iterative steps in collimating, using the XLK AC and also the BLACKCAT ™ Cheshire exuivalent and the HotSpot center marker:

To quote Jason's captions:

You may note that in the last two rows, collimation errors are so small that all reflections but P have vanished, as seen through the centered pupil (left column).

Some finishing words:

Today, the autocollimator has gained wide popularity, no doubt through better understanding of its operation ( - and not least, by using the "carefully decollimated protocol" or CDP, first suggested here, and tried and proven by Vic Menard. While the autocollimator can give a very good collimation, is it better - more accurate, and/or simpler to use, than for instance, the Barlowed Laser?

No doubt both methods can bring collimation to well within tolerances, so it's a matter of personal preference.

Nils Olof Carlin

Latest update July 21, 2010