Interferometers – (second part)

Interferometers – (second part)


In this example shown (Fig 1), the mirror is a 200mm diameter F/5 parabolic mirror with a Ross lens tested at the centre of curvature. The basic optical layout of the mirror with Ross lens and focus position is shown at bottom left with the “Prescription” for the optical system above it.
We arbitrarily placed the Ross lens 300mm in front of the interferometer on the optical axis (Z axis) and then optimised the Ross lens radii until it achieved a value better than 1/20λ PV. The result is the picture at bottom right in Fig 1.
Once the system was set up and optimised, we deliberately moved the Ross lens 3 mm along the axis.
A small shift of the interferometer position was then needed to allow the out going and return beams to coincide in the interferometer. The result is Fig 2 adjacent.

This suggested a error a little worse than 1/3λ. Just to make this clear – This means if an Interferometer was used to test a perfect mirror with the Ross lens 3mm out of position along the axis, it would appear that the mirror had a PV of 1/3λ when it was in fact a perfect mirror!
1mm movement would suggest the mirror was about PV 1/10λ out.
We then moved the Ross lens back to its correct position at 300mm along the axis and then shifted it off the optical axis (Y axis) by 3mm.
The result is Fig 3 adjacent
This position gives 1 1/2 fringes of astigmatism,
So a mirror tested at these settings would appear to be PV 1.5λ out which is a massive amount in an astronomical mirror!
Scaling that back to get an accuracy of 1/10λ requires the Ross lens to be positioned within 0.2mm of the optical axis of the system.
We would find it very informative if an interferometer tester can explain how he positions the Ross lens on the optical axis to the accuracy needed – And then how he determines where along the optical axis to position it. But if he cannot position the lens accurately, he cannot obtain accurate results from the interferometer testing.
This example is just about positioning the single Ross lens in relation to the mirror. We have not considered the alignment to the body of the interferometer or the placement of the extra lens used to convert from a flat to a spherical Wavefront. If one lens and mirror cannot be positioned properly, then how can the other optical items be correctly aligned?

We know of yet another method being used: the Ross correction lens can be left out and the fringes created from the spherical aberration can be used instead of tilting the reference flat. Although we are aware that this method has been used on some mirrors, a 20″ F/4 would show about 54 fringes from spherical, and straightaway it is difficult to see how accuracies within a typical RMS of 1/35λ could ever be achieved.
You are be asking the interferometer plus all the lenses in the set-up to have a total accuracy to better than 1/10 of a percent! The computer has the job of subtracting the fringes seen in this set-up from what a perfect mirror would give and use any remainder to calculate the error on the mirror surface or Wavefront. This will only leave extremely subtle shades of grey to calculate error from. Spherical aberration produces circular fringes and they are not regularly spaced: there will be a big gap in the centre of the mirror and the fringes would crowd together towards the edge of the mirror. This will give poor accuracy in the centre within the first fringe. There will also be accuracy issues at the edge due to the fringes being extremely close together.

The picture adjacent is of the idealised fringes from a 6″ F/8 generated by computer. If you look carefully at the inner black fringe in this picture, you will note a well defined outer edge with a sharp transition between black and white, – but the inner edge of the fringe is less distinct with more and longer shades of grey.
This would indicate that the real optical centre of the fringe may be nearer its outer edge than in the physical centre of the black band, where you may be tempted to place it. So where is the exact optical centre of this fringe? There are no fringes in the centre of this picture, so the computer may try to work solely with the level of grey. If the light source does not produce even illumination across the width of the picture or the CCD response is not perfect, the Interferometer results will suggest there is a bump in the centre of the mirror.ù
To counter this, the mirror, or the internal reference in the Interferometer, can be tipped slightly off axis again. The picture adjacent is the idealised computer generated fringes from the 6″ F/8 tipped 2.5 waves, (5 fringes), and set at “best fit”.
This does give some fringes across the centre of the mirror but they are now curved and of variable width instead of straight. Now that they are curved and with different densities of grey each side, It is again difficult to work out where the centre of the fringes are.
Another thing that intrigues us about Interferometer testing is we often see test results suggesting a mirror has astigmatism, – but it does not appear that the very simple test for astigmatism using an eyepiece is carried out to confirm the astigmatism exists. Astigmatism is very rare in a professionally made mirror. If this simple test confirmed that no astigmatism exists, then any results from an interferometer suggesting otherwise are wrong and probably due to an incorrect set-up.

We believe this simple test for astigmatism is crucial. We believe a mirror should always be proved free of astigmatism before any other bench testing is carried out. This test is always carried out on our mirrors before they go for the Double Pass Null. Then we have confidence there is no astigmatism in the mirror. We believe this simple  test should also be applied before testing with an interferometer.

The strengths of Interferometer Testing are:-

  • Able to sample tens or hundreds of points on the mirror and calculate RMS & Strehl ratio quickly.
  • Results can be highly customised. It can produce highly impressive reports containing pictures, graphs, 3D pictograms and tables of figures.

The attractive reports inspire confidence in the testing and imply it has an impressive accuracy.

(We don’t agree about the impressive accuracy of course!)

If you are considering an independent Interferometer test, you may wish to clarify the following details first:-

  • The Model of Interferometer to be used
  • Confirm the method [Fringe Analysis?] [Double Pass Optics?]
  • A full diagram of the optical path used for the tests
  • Details of any additional optics used; Including Lens details; F/ ratios; focal lengths; Distances & Spacing
  • Method of setting up and use of the equipment
  • Any ancillary equipment or programmes used to process and produce the results.

If you have problems obtaining these basic details, then you might choose to look elsewhere to get your mirror tested? You might also ask for the test on your mirror to be done twice – Repeated – with the mirror rotated through 90 degrees.

“Dead-Give-Aways” Interferometer set-up problems.

If you do get an interferometer test done, there are some things you can look for in the interferometer report that indicate it was not performed adequately. All below are examples seen in Interferometer test reports.
The first thing is do the results look central? In the example adjacent we suggest its very obvious something is off-centre.
Taken by itself, this particular example would not have affected the main interferometer test results as they could have been derived from another set-up, – but it should perhaps be worrying to you that if this one is off-centre, then the other tests might have been.

Much more serious is obvious coma in the interferometer results. The test report you receive will usually have a picture of the wave-front as seen by the interferometer. An example is the picture showing green fringes adjacent.
If you see “S” bending in the fringes, (like in this example), the report probably wont explain this is coma due to errors in the interferometer set-up. One or more of the optical elements involved was not quite on axis or was at a small angle to the axis.

The tester is expecting the interferometer software to note the errors in the set-up and subtract them from the final result. For this test we show the final result adjacent to the fringes. If you consider the surface underneath the black line superimposed, the shape of the mirror under the line is again an “S” curve. This is still showing coma and is strong evidence the interferometer software had not correctly isolated and subtracted the errors in the set-up.
How are we sure its errors in the set-up and not in the mirror? – That’s dead easy – we can’t make mirrors with shapes like this!
The process of grinding, polishing and figuring a mirror is basically rotary and is symmetrical. Astigmatism is removed by the mirror maker in the early part of the process. Therefore any errors in the mirror will nearly always be symmetrical. Either not quite the correct curve on the mirror or perhaps some circular zones.
ERR – No zones on our mirrors, – thank you very much!
So seeing an “S” curve on the fringes and then on the mirror profile is a “Dead-Give-away” that the interferometer set-up was incorrect and the software was unable to process the results correctly. We have just mentioned coma so far. there is probably also some astigmatism in the set-up plus some spherical aberration caused by an incorrect distance between the various optical elements. then there is the real error on the mirror. We suggest there are too many terms for it to properly separate out the real error. This test reported the PV error on the mirror was 0.5λ.
But its certainly a pair of very nice pictures for the test report. There can be no argument it looks extremely professional – That might help convince you the test is accurate?

Now you know what to look for, – here is another example. This one is a lot better than the previous example as the Interferometer fringes are nearly straight across the majority of the mirror.
But look at the top and bottom of the fringes. At the top they giving a tell-tale bend right and at the bottom they are bending left. Its the “S” curve again! – Coma from incorrect set-up.
The matching mirror surface profile is next to the fringes and in this case the “S” curve can be made out running nearly East to West on the surface under the black line superimposed.
The error from poor set-up is still the dominant error in this test. Even with just the small bends at top and bottom of the fringes to suggest coma, it’s still impossible to derive a meaningful figure for the real error on the mirror surface. This test reported the PV error on the mirror was 0.14λ.
Certainly we can speak for ourselves, – and probably a lot of other professional mirror makers, – that any error remaining in our mirrors is predominantly symmetrical. Probably due to not quite having the correct curve. (But still within specification!) Any signs of asymmetry in Interferometer test results point to difficulties in the Interferometer set-up.

Next is a mirror test result that has the mirror looking more like a bent bottle top! Unless someone can give us a better explanation we believe its an example of where a dark fringe lies along the edge of the mirror and the interferometer software can’t interpret it accurately. In this case its happening on opposite edges at east and west.
This is mentioned as a problem earlier on this page On the edge of the mirror the interferometer may have only part of a dark fringe with no “white” (actually green in these pictures), area outside it to use as a reference when fixing the fringe centre. We believe the interferometer software cannot reliably find the centre of the fringe at these positions so introduces an error. This particular test reported the PV error on the mirror was PV 0.2λ.
Since this is virtually all due to the peaks at the east and west edges, and they are probably errors in locating the fringes, the mirror was really extremely good and certainly better than 0.1λ
We will repeat yet again that any errors remaining in a professionally made mirror ought to be symmetrical. Any sign of asymmetry in interferometer reports ought to be a “Dead-give-away” that there were problems in the Interferometer set-up or the software interpreting the fringes.

Interferometers are very good at producing impressive reports. Unfortunately that can inspire over-confidence in their results.
We suggest that Interferometers are completely fine where the accuracy required is above a wavelength. However Astronomical mirrors are large in comparison to interferometer optics and the accuracy required is a small fraction of a wavelength. In this area test results from Interferometers become highly dependant on the equipment quality and the skill of the operator. In practice there is no easy way to set-up an Interferometer test so no way of avoiding errors creeping into the set-up. Undetected errors in the set-up appear in the results as errors in the mirror under test.
Even if the set-up is perfect and the tester is absolutely superb, there is a limit on accuracy imposed by the extra optics in the Interferometer. Results will be less accurate than those obtained with a Double Pass Null.
Along with most other professional manufacturers of astronomical mirrors, we believe the most accurate method of testing a mirror remains the conventional Double Pass Null Test.

Oldham Optical

The Airy Disc

It can be extremely surprising and annoying to find out that the big parabolic mirror you have just paid a lot of money for cannot focus all the incoming light down to an infinitesimally small point, but instead puts it into a finite sized disc with some faint rings around it.
You have been told the Airy Disc cannot be avoided and that it’s due to diffraction and the wave nature of light, but every explanation you look up soon disappears off into the deeper realms of mathematics. It all seems a bit of a fiddle!
Well – I am afraid a lot of the mathematics cannot be avoided if you need to go into the subject deeply, but it is possible to get a working knowledge of what’s happening without too much of the maths and with just a bit of thinking in the bath and a bit of work on the kitchen table with a ruler and compass.
Perhaps the easiest way to start to visualise what’s happening with your parabolic mirror is by considering the wavefronts of light coming in from a distant star. The star is so far away that the light rays approaching can be considered parallel, and the approaching wavefronts can be considered flat plane lines, across the incoming parallel rays.
If you live at the seaside, stop and look at a series of waves marching in towards the beach. If the tide is well up and they hit a concrete sea wall, – you will often see them reflect back. If the sea wall happens to be curved, you may even see some of the reflected waves converge, (focus), towards a point.
The Airy Disc –

An Explanation of What it is and Why You Can’t Avoid it

Unfortunately, not many sea walls are designed with the perfect parabolic shape to be an example of how a parabolic mirror works but any concave curve should show some focussing effect, enough for you to get an idea of what’s happening.
(Don’t lean over the railings too far and if anyone starts looking at you strangely – its probably time to move on and get an ice cream!)
So getting back to light and a parabolic mirror – The flat wavefronts march in to the mirror like waves in the sea and are reflected. The parabolic shape of the mirror converts the incoming flat wavefronts into reflected spherical wavefronts. These spherical wavefronts converge at the focal point of the mirror. See the diagram below.
If the mirror was perfect, light was not comprised of waves and diffraction did not exist, the focal point where all the light converged would be infinitesimally small.
Sorry! – It’s just not like that

Diffraction is something that happens at edges. When something like light meets the edge of a solid object, some of the energy tries to bend round it.
The usual diagram seen in Physics textbooks to illustrate Diffraction is of flat wavefronts approaching a screen with holes in it. The holes are small and close to the wavelength of the light used. The light passes through the holes and then part of it tries to bend around the edges of the holes as it emerges.
If the holes are small and not many wavelengths of light wide, then the resulting wavefront from each is spherical. It appears as if the light is being generated from point sources situated at the holes themselves, rather than the actual source which is somewhere behind the screen
The really interesting thing is where you have two or more holes all acting as point sources, the light from each hole will interfere with each other. That is what happens with a Parabolic mirror to form the Airy Disc and the rings.
There is an experiment you can do in the bathroom at home that illustrates the same wave effect, but note this health warning first – it might not be a good idea to let your wife find out you are carrying out experiments of this nature!
So sneak into the bathroom armed with blocks of wood and float them in the bath. They should be spaced with a small gap between them. Make waves on one side and watch how they propagate through the hole.
If you don’t think you dare try this in the bath, you can do the same thing just as well in the garden pond with a couple of rocks, but do not let the neighbours see you! They might tell the wife!

In real life, the reflected light from the edge of the mirror merges with the rest of the wavefront reflecting from the main surface of the mirror.
Most of the combined wavefront is spherical from the main body of the mirror but bends back in the opposite direction towards the edges.
It is the departures at the edges that form the rings around the central Airy Disc.
Although it sounds very complicated, its actually very simple to demonstrate what is happening just by sitting down at the kitchen table and drawing out the wavefronts on a piece of paper and a compass.

Start by drawing a mirror say 2 “ in diameter. Then with a compass you are going to draw out concentric arcs from each rim of the mirror to represent the wavefronts as in the diagram adjacent. Each arc must be regularly spaced as it represents a wavelength
So draw say 10 arcs from each rim of the mirror starting at say 2″ radius, and at every ½” out to say 7½”. The arcs from each rim will intersect.
If these pencil arcs were waves of light they would interfere with each other. Where two arcs cross there would be a bright patch of light and where one arc is half way between two other arcs from the second rim, they would cancel out and give a dark patch.
If you then look at what you have drawn you will see you can draw a series of straight lines through the intersections. All lines begin from roughly the centre of the mirror. These lines represent the bright patches of light.
They are the black dotted lines in the diagram, with the central bright spot shown in a heavy black line.
Now steadily go through the drawing and put a mark where one line from one rim is half way between two lines from the other rim. Once finished the marks should lie on a second series of lines that again start at roughly the centre of the mirror and run out through the marks. This series of lines is the dark patches that exist between the Airy cisk and the rings.
The first two defining the Airy Disc are the blue dotted lines in the diagram.

Although on flat plain paper these are drawn as lines, on a three dimensional parabolic mirror they represent cones of light. Where these cones intersect the focal plane of your telescope they will appear as a bright central disc with rings around it.
The central disc is of course the Airy Disc. In practice, when looking through a telescope the rings rapidly get faint and it is usually only the first ring and perhaps the second that can be seen in practice.
The angle of the disc and of the first ring is illustrated on the diagram above. The first Airy Ring is easy to work out. It is the first set of bright intersections away from the central spot. The Airy Disc itself is not quite as easy to see at first.
The accepted convention for the size of the Airy Disc is to measure to the centre of the first dark patches either side of the central spot. The diagram should give you the lead.
Instead of using the term “angle to the disc”, the term more commonly used in astronomical circles is “angular diameter”. So please note that when the rest of this document uses “angular diameter”, it means the angle to the Airy Disc (or rings), as described above.
If you repeat the drawing with different diameter mirrors, say 1” and 3”, while still using ½” rings to represent the same wavelength, you will find that as you increase the diameter of the mirror, the angular diameters of the disc and the ring reduce. Since the angular diameter reduces, the physical diameter of the disc and rings also reduce.
Although this illustration is just something done on the kitchen table, It is exactly the same mathematics that apply with light to form the real Airy disc and Rings. The angular diameter of the disc and rings is calculated with exactly the same rules. It follows that the Airy Disc gets smaller as the diameter of the mirror increases.
Before the people who really understand mathematics jump in and complain that the intersections are not really on straight lines, they are on hyperbolic curves, it is agreed that they really are on slight curves rather than straight lines! The lines come in from infinity and do bend very slightly as they approach the mirror. So evidence of a slight curve is likely to be noticed on the kitchen table using wavelengths of ½” with a 2″ mirror.
However light has a wavelength of about 500nM, so the line for the first Airy Ring will have come in from infinity and will bend out of line near the mirror a maximum of ½ a wavelength, which is about 250nM. It is submitted that coming all the way from infinity and only bending 250nM out of line is still pretty damn straight!
In any case,  the accepted formulas used for working out Airy Disc size and angular diameter assume the lines are straight, so no reason this description cannot be allowed to do the same!
The description above only covers light from the rim of the mirror – so what about the light from the rest of the mirror? Don’t forget the rest of the mirror surface is pouring light into the central spot. The very simple explanation is that this light merges with the light scattered from the rim of the mirror to give the bright Airy Disc.ù
The only practical thing a telescope constructor can do to make the Airy Disc smaller is to make the mirror diameter larger. This makes the angular diameter of the Airy Disc smaller. The angle is fixed for any one diameter of mirror. Once the angle is fixed, the physical size of the Airy Disc is determined solely by the focal length of the mirror.
A common formula used for working out the size of the Airy disc uses the Focal ratio, which includes both the diameter of the mirror and the focal length in one term.
One formula for the Airy Disc is:

  • D = 2.43932 x λ x Focal Ratio
  • D = Diameter of Airy Disk in mm
  • λ = Wave Length in mm (e.g. 546nM = 0.000546mm)
  • (If Focal ratio = F/4 & 546nM used then D = 0.00533mm)
  • One formula for the diameter of the Airy Disc is:
  •  A = 7200(Arc Tan(1.21966 x λ /d))
  • A = Angular Diameter of the Airy Disk in Arc Seconds
  • d = Diameter of Mirror in mm

Ok! So you now know what the Airy Disc is! But from the description above you should be thinking there’s a easy way to get rid of the thing!
Surely if it’s just the light at the edge of the mirror that causes diffraction, then why not just mask off the edge of the mirror with a cardboard ring? This will stop light hitting the edge of the mirror. Wont this get rid of the Airy Disc?
Sorry – this does not work! You now get diffraction from the edge of the mask instead! What you have done by masking the mirror edge is add what is called an “Aperture Stop” to your optical system.
By doing so you slightly raise the focal ratio of the equation and the Airy Disc actually gets bigger.
This also explains why in your telescope you need to keep the spider holding the elliptical flat or secondary mirror as small and unobtrusive as practical. Any obstruction causes additional diffraction.


Finally, – here is the killer – let’s suppose you could magically mask off the edge of the mirror without causing extra diffraction. You have managed to limit the beam of light falling on the mirror to less than the mirror diameter so the edge of the mirror is not illuminated. You must really think you have the problem cracked now?
Sorry – This still does not work! The edge of the beam of light falling on the mirror surface is itself a discontinuity. When it strikes the mirror and reflects it behaves exactly as if it was a real physical edge.
There is diffraction from the edge of the beam exactly the same as if it was the physical rim of the mirror. For this situation there is no easy and simple “kitchen sink” explanation. This is where I “cop out” and say this is due to the wave nature of light and you do need to start digging into the physics text books if you want to understand it better.
Exactly the same thing happens if you paint the outer edge of your mirror with black paint. All this does is cause diffraction to occur where the black paint finishes and the reflective surface starts. All you have done with the paint is reduce the diameter of the mirror. The result is that the Airy Disc gets bigger.
The message is that you cannot avoid the Airy Disc.
However you can make it smaller: You can either increase the diameter of the mirror, which reduces the angular diameter of the Airy Disc, or reduce the focal length of the mirror, which reduces the physical size of the Airy Disc.
The Airy disc size is suddenly becoming more important to amateur astronomers due to the recent improvements in digital cameras where the CCD sensor pixel size has now reduced to the point where it is comparable or smaller than the Airy Disc size. In future, telescope mirror diameter and focal length may have to be chosen to better match the CCD pixel size?
To close, it is possible you could completely avoid the Airy Disc by using an infinitely large mirror – but I would respectfully suggest a long talk with your wife or perhaps your doctor before trying to order one of these.