*This page follows on from my more general webpage on Atmospheric Dispersion Correctors which I recommend you read first*.

*First published on 22/12/2016*

*Modified on 26/12/2016*

## Introduction

Although ADCs can markedly reduce atmospheric dispersion, so improving image resolution, when used at larger adjustment settings on objects at lower altitudes, they can introduce significant optical distortions (optical aberrations). Although aberrations will be worst when the object is low in the sky and seeing effects will invariably be worse too, it is useful to understand their limitations and to understand when these aberrations exceed the benefits of using an ADC, enabling you to get the best out of them.

Information on optical aberrations introduced by ADCs is impossible to find online and is usually limited to crude statements like ‘they must be used above f20 or f15 or they will introduce astigmatism’. To help understand the performance of the common simple two-prism ADCs, such as those sold by ZWO, Pierro Astro and ASH, and to fully understand when this performance starts to break down, I worked with UK optical engineer, Es Reid, who used Zemax optical modelling software to determine the magnitude of the aberrations under different conditions for two example sizes of telescope.

The insight gained in this analysis enabled me to make some deductions about their performance which may be useful to owners of all sizes of telescope who use ADCs. The section below summarises these main findings and this is followed by the more detailed analysis.

## Main Findings

- Aberrations produced by an ADC only become significant (larger than the airy disc) at object altitudes below about 30°
- At best focus, the aberration pattern produced by the ADC is generally triangular in form and gets worse the shorter the focal ratio (steeper the cone of light entering the prism) and the larger the prism angle.
- Fortunately, you only need to worry about the
*altitude*of the object itself and not the individual prism/focal ratio settings of the ADC when concerning yourself with the aberrations the ADC might introduce. At any one altitude, aberration changes arising from changing the focal ratio tend to be cancelled out by the aberration changes in the other direction from the corresponding prism changes that will be needed to maintain dispersion correction for that altitude. Hence, it is rather misleading to simply say, for example, don’t use them at f15 or less or don’t have a prism angle of 6° or more.

- The win/lose balance of using an ADC on a particular telescope at low altitude depends on the relative sizes of the linear atmospheric dispersion spectrum if you don’t use an ADC, versus the size of the aberration pattern if you do use one. The former depends on the wavelength and bandwidth of the filter being used, the focal length of the scope being used whilst the latter depends on prism angle, f-ratio and ADC to focal plane distance. For a 457mm scope;
*At 25° or higher the dispersion is worse than the aberration for all filters so use the ADC**At 20° the dispersion is worse than the aberration for the L, G and B and possibly the 642nm but for the red the aberration is probably worse than the dispersion**At 15° the dispersion is worse than the aberration for the L, and the B but for the R, G and 642nm the aberration is probably worse than the dispersion**At 10° the dispersion is worse than the aberration for the L only. For other colours the dispersion is big but the aberration from using an ADC is bigger. There may be an argument here to using an ADC and only partially correct for the dispersion to keep the aberration under control but give some benefit.*

- Smaller scopes with shorter focal lengths produce shorter dispersion lengths for a given f-ratio and altitude of object which means less correcting power (prism angle) is needed in the ADC. This greatly reduces the aberrations introduced. As a rule of thumb if you halve the size of the scope you can halve the altitude for the same level of aberration relative to the size of the airy disc. You can apply this to the altitude values in the previous section which will all be halved in a scope of half the aperture.

- Based on the above relationship between the aperture and the altitude for a given aberration you can make a approximate table of altitudes at which the aberration exceeds the dispersion for different filters. I have included a small scope (115mm to illustrate the linearity of the trend of acceptability with altitude rather than because this will be directly useful to imagers);

Note that for one-shot colour cameras which have wider bandwidths for each colour then the dispersion effects will be worse. This means there will be benefit at lower altitudes before the aberrations dominate over the (larger) dispersion compared to a mono camera with colour filters.

- The prisms in an ADC cause a vertical displacement of the object when applying the correction and you need to tilt the scope to bring the object back to the camera chip centre. The offset from the optical axis and the resultant miscollimation that this brings about, however, can be found to cause a negligible amount of extra aberration.

- It is worth minimising the ADC to barlow distance to maximise the ADC to focal plane distance. This will maximise the corrective power of the ADC without needing to increase the prism angle so optimising for aberrations.

#### Aberration Modelling Method

As a vehicle for exploring the aberrations, optical modelling was first based on an 18inch (457mm) reflecting telescope with a parabolic mirror (ie a Newtonian reflector). BK-7 glass prisms were used in the ADC and the ADC was positioned after the Barlow lens. Most commercial ADCs have a (combined) prism deviation angle of either 0° to 4° or 0° to 5° so to mimic this range of corrective powers three different deviation angles were modelled in Zemax; 2°, 4°, and 6°. In addition, a range of different Barlow magnifications were used, yielding four different effective focal ratios (f10, f15, f20, f30) together with two different example ADC to focal plane separations (70mm and 105mm).

The aberration spot plots for the different conditions are shown in figure 2. The following things should be noted when reading this figure;

- Top set is for a 70mm ADC to focal plane distance whereas bottom set is for a 105mm ADC focal plane distance

- The aberration spot plots are shown in three groups of four with all in each group having the same prism deviation angle of 2°, 4° or 6°

- Within each prism group are the same four focal ratios of f10, f15, f20 and f30

- For each prism deviation angle and f-ratio combination the focus was chosen to make the spot plot the roundest and tightest ie best focus. Note- A perfect aberration-free ADC would give a spot plot that would be a single spot

- Dispersion if uncorrected would be in the vertical direction with the light spread into a vertical spectrum. The ADC corrects for this eliminating this vertical spread but introducing aberrations which are then symmetric about the vertical axis (see figure 1).

- The calculations of aberration all assume perfect dispersion correction by the ADC which will not be the case as the dispersion curve of the BK7 does not perfectly match that of the atmosphere especially in the infra red part of the spectrum

- There is a 100um scale on the LHS of both sets to measure the magnitude of the aberration against

- The star’s Airy Disc (AD) is shown for each combination as a circle. For the ADC to be diffraction limited for that combination an optician would say that the spot plot should lie entirely within the Airy Disc

- Underneath each line of aberration figures you see the altitude at which a star’s dispersion matches the dispersion (but of the opposite direction) introduced by the ADC prism. At the altitude given the object’s dispersion is properly corrected by the ADC.

There is a lot of data shown in figure 2 so here are some more notes to better help understand what is shown here;

- The 105mm set represents 1.5x the ADC to focal plane distance of the 70mm set. Increasing this distance effectively increases the correcting power of the ADC without increasing the prism angle and it’s pretty much linear so 1.5x is like increasing the prism angle by 1.5x. As an example of this see at f30, a prism angle of 6° for the 70mm corrects for an object at 20°. Look now at a prism angle of 4° (1.5x smaller) and 105mm (1.5x larger) and this corrects at an altitude of 20° too.

- Higher prism angles and larger ADC to imaging plane distances increase the ADC power and allow correction to be achieved for lower altitudes of the object where the atmospheric dispersion will be greater

- The larger the focal ratio the larger the Airy Disc due to the higher magnification that arises from the longer focal length. The ADC to focal plane distance has no bearing on the AD size but in reality moving away from the ADC generally also increases the Barlow to focal plane distance which can change the Barlow power. In these worked examples we assume the Barlow to imaging plane distance is constant when going from an ADC to imaging plane distance of 70mm to that at 105mm hence the AD size is unchanged. Altering the ADC to imaging plane distance without changing the Barlow to focal plane distance (don’t forget the ADC is after the Barlow) would be a bit hard to achieve in reality but this restriction helps keep things simple here.

- The explanation of why the altitude value given increases with increasing f-ratio within each group (same ADC distance and prism angle) is tricky to understand. Increasing the focal ratio of the scope by changing the Barlow etc increases the magnification of the scope but has no effect on the amount of corrective dispersion produced by the ADC. However, because the magnification is higher at higher focal ratios this means that the linear dispersion spectrum of the star created by atmospheric dispersion is longer as it is magnified more at higher focal ratios than lower ones. Now, in order to match the fixed size of the corrective dispersion length of the ADC at 70mm or 105mm separation (in terms of microns of length) the angular atmospheric dispersion has to be less at higher magnification than at lower so that the length of the star’s spectrum ends up the same and can be corrected by the ADC. Overall this means that at higher f-ratios the ADC cannot correct for such low altitudes as at lower f-ratios.

- At the bottom of each set you see groups of coloured vertical lines. These are the amounts of linear dispersion, in microns, introduced by each prism angle and ADC to imaging-plane distance set. The lengths depend on the wavelength and the wavelength range so vary for different colour filters. The length of the lines are scaled to match the 100um bar and the aberration and airy disc plots. Each line is for a different filter and see how the dispersion increases as you move into the blue and also as you widen the acceptance bandwidth;
- L=Luminosity filter, Astronomik; acceptance band from 390nm to 690nm
- 642nm=Astronomik Planet Pro; acceptance band from 642nm to 842nm
- R=Red filter, Astronomik; band from 580nm to 665nm
- G=Green filter, Astronomik; band from 490nm to 590nm
- B=Blue filter, Astronomik; band from 390nm to 500nm

The lengths of the lines are related to the dispersive power of the ADC in each set. So the lines in the 4° group are twice that of the 2° group and the lines in the 105mm set are 1.5x those in the 70mm set. Also note that the ratio of the size of the airy disc to the length of the lines is constant for a given altitude (say 20°) regardless of the combination of prism angle/ADC distance/f-ratio that the altitude corresponds too this is because both the airy disc size and dispersion are a fixed angular length.

- It is of key importance to understand that the linear dispersion represented by the vertical coloured lines shows you the magnitude of the dispersion aberration that is introduced by the atmosphere for that group. The aberration shows the spot plot assuming there is no dispersion and it is all removed by the ADC so, if you like, it gives an idea of the vertical smearing that would be applied to the Airy Disc is the ADC was not there. If the ADC didn’t correct for it then you would have vertical smearing of every point of the image by the amount represented by the vertical line corresponding to the filter you are using. With the ADC in place and working correctly the vertical smearing isn’t present but the aberration you see in the spot plots replaces it.

## Key Findings from Figure 2

1. The degree of aberration introduced by the prisms of the ADC is determined by two main things- the correcting power of the ADC and the degree of convergence of the light entering the ADC prisms (this is just determined by the f-ratio of the telescope/Barlow combination used);

- For a given f-ratio, as the dispersion correcting power increases (prism angle increases and/or ADC to imaging plane distance increases) then the aberrations with respect to the size of the airy disc get worse.
- For a given correcting power as the f-ratio decreases the aberrations with respect to the airy disc get worse, as the angle of convergence of the light gets larger.

Increasing the f-ratio reduces the angle of convergence reducing aberrations, however, the higher magnification that the increased f-ratio gives, leads to a wider dispersion length for the object. This then needs more prism power to correct this which in turn increases the amount of aberration. Overall the data shows that for a given altitude of object these two characteristics tend to essentially cancel each other out and the aberrations stay about the same as you change the f-ratio (and of course also alter the prism angle to achieve dispersion correction for that fixed altitude).

As a consequence of the above, within each set (70mm or 105mm), the size of the aberration pattern compared to the size of the airy disc is primarily related to the altitude of the object rather than the specific f-ratio or prism angle. The simplistic statement that you shouldn’t have an ADC prism angle greater than 6° or an f-ratio less than f15 is just misleading.

To illustrate this dependence of the amount of aberration on the altitude, look at the 70mm set and compare the following;

- For 29°-30° altitude= 2°/f15; 4°/f30; 6°/f30. All about the same and aberration figure just smaller than airy disc
- For 19°-20° altitude= 2°/f10; 4°/f20; 6°/f30. All about the same and aberration figure just bigger than airy disc
- For 13°-15° altitude= 4°/f15; 6°/f20. Both similar and aberration figure ~3x dia of airy disc

This rule holds within the 105mm set too;

- For 20° altitude= 2°/f15; 4°/f30. Both about the same and aberration figure similar size to airy disc
- For 13° altitude= 2°/f10; 4°/f20; 6°/f30. Both similar and aberration figure ~2x dia of airy disc
- For 9°-10° altitude= 4°/f15; 6°/f20. Both similar and aberration figure ~4-5x dia of airy disc

It seems that for a given altitude the aberration is a bit less for the 105mm set than the 70mm set as increasing the dispersive power occurs without the need to increase the prism angle. For a given Barlow to imaging plane distance (fixed magnification and focal ratio) it is worth minimising the Barlow to ADC distance to make the most of this benefit as this will help keep aberrations minimised.

2. Whether it is an advantage to use your ADC for an object at a particular altitude isn’t a question of whether the aberration pattern is smaller than the airy disc, it is a matter of determining if the aberration introduced by the ADC is greater than the benefit of removing the dispersion. To find this out just compare the vertical bar length for the filter you are using against the size of the aberration pattern. Please do also take into account that the atmospheric dispersion is purely smearing in a vertical direction whereas the ADC induced aberrations are both in the vertical and the horizontal directions.

Some simple rules can be formulated based on the figure and are applicable to scopes of the same aperture;

- At 25° or higher the dispersion is worse than the aberration for all filters so use the ADC
- At 20° the dispersion is worse than the aberration for the L, G and B and possibly the 642nm but for the red the aberration is probably worse than the dispersion
- At 15° the dispersion is worse than the aberration for the L, and the B but for the R, G and 642nm the aberration is probably worse than the dispersion
- At 10° the dispersion is worse than the aberration for the L only. For other colours the dispersion is big but the aberration from using an ADC is bigger. There may be an argument here to using an ADC and only partially correct for the dispersion to keep the aberration under control but give some benefit.

## Effect of Lateral Offset

One consequence of increasing the prism angle from zero in an ADC is that the image will move laterally from the centre of the field in the vertical direction. For big adjustments this can easily push the image right outside of the imaging chip area. Tilting the scope moves the image back to the chip centre but now the object is effectively off-axis and the scope is slightly miscollimated. In the previous two figures the Zemax calc assume that this tilt has been applied bring the image back to the chip centre – this is just how people normally use their ADCs.

To see if the image was much better before the tilt was applied and if a lot of the aberration was due to this unintentional miscollimation, Es calculated the aberration spot plot for the image offset from the chip centre and now sitting on the optical axis. The figure below compares the recentred/tilted set for a 70mm ADC to imaging plane distance to the same where no tilt has been applied and the image if offset from the chip centre. The tilt applied for each combination is indicated for the top set as is the offset calculated for each of the three prism angles.

Some slight benefit in not tilting to recentre the image on the chip is seen at f10 but at higher f-ratios there is essentially no difference in the degree of aberration between centred and offset.

A Newtonian scope performs very well on-axis but compared to other telescope designs it suffers quite badly from astigmatism and coma off-axis. The minimal difference on and off-axis seen in the figure above would suggest that other scope designs (Dall Kirkham, RC, SCT, refractor) most of which are better than a Newtonian off axis would also be little different on and off-axis and would in fact perform similarly to the Newtonian case given the same f-ratio.

This is a key finding as this implies that the conclusions in this webpage are applicable to most telescopes of this aperture regardless of type.

## Effects of Smaller Aperture

To see the dependence of the aberrations on the size of the telescope Es ran the modelling for a 235mm scope for the same 70mm separation; 2°/4°/6° prism deviations; and set of focal ratios, f10, f15, f20 and f30. The plot below compares the 457mm result for no tilt with the same no tilt set for the smaller 235mm aperture.

Amazingly the aberration spot plot graphic looks virtually identical for the two different aperture telescopes under the same conditions. This is good news as it allows one to say what the aberrations will be like for any aperture scope operating in this same range of prism angles and f-ratios- it will look just like above. It looks the same because as we have said previously the severity of the aberrations solely depends on the angle of convergence of the light into the prism (ie the f-ratio) and the correcting power of the prism set up (prism angle and distance between prism and focal plane). As these are the same in both cases above then the size and shape of the aberration pattern is just the same.

It may also be a surprise to you that the Airy Disc diameter is the same for both apertures. Surely the Airy disc gets bigger with reducing aperture doesn’t it? Well angularly it does, but in terms of linear diameter at the focal plane it stays the same for any given f-ratio. Let’s go from a 457mm scope to a 235mm scope both operating at f30. In doing so the aperture drops to 51% of the original, making the Airy Disc 51% larger angularly. However, the focal length has dropped to 51% of the original too and this makes the Airy Disc 51% smaller as linear magnification is just a function of focal length. Thus the linear size of the Airy disc in microns stays the same in both cases. Having said this, do remember that the size of a planet will drop with decreasing aperture for the same f-ratio so compared to the size of a planet the airy disc will become comparatively larger in a smaller scope.

What does change in changing the aperture is the altitude of the object where the dispersion is fully corrected by the ADC. For the scope that is half the aperture, the altitudes at which the dispersion is properly corrected are generally about halved too. The explanation for this relates to the length of the dispersion spectrum produced for the two scopes at the same f-ratio. For a given altitude of object the smaller scope will produce a dispersion length half the length of the bigger scope. The ADC will overcorrect in this case. If the object drops in the sky the dispersion length will increase. Interestingly at altitudes of 30° or less the atmospheric dispersion doubles every time you halve the altitude. So halving the altitude doubles the dispersion length and gets the dispersion length for the small scope back up to that produced by the bigger scope where both are then corrected.

Overall the smaller aperture is much less affected by ADC aberrations for a given altitude. Lets pick an example to illustrate this; compare the aberration pattern for an object of 10° altitude. For the 235mm scope the aberration pattern is slightly larger than the airy disc whereas for the 457mm scope the aberration pattern much larger than the airy disc. In the 457mm scope the angular size of a planet is twice that compared to the 235mm scope but the aberration spot plot is way bigger than twice the size. Overall the bigger the scope the worse the aberrations compared to the object!

## Putting the ADC before the Barlow

Although it is generally recommended to have the ADC after the Barlow so that the cone angle of the incoming light is much narrower, some users with longer inherent focal length scopes such as SCTs have reported good results with the ADC before the Barlow. Doing this impacts the aberrations in several ways;

- It increases the cone angle of the light entering the ADC leading to worse aberration effects

- It increases the distance between ADC prisms and the imaging plane so that a given angular convergence of the blue and red images acts over a much greater distance increasing the ADCs correcting ability. This in turn allows the prism angle to be significantly reduced leading to a reduction in aberrations (this gives extra headroom and may allow proper dispersion correction in cases where with the ADC in the more usual position after the Barlow would not enable correction to be achieved)

- The longer distance between ADC and imaging plane will increase the linear size of any aberrations present just because the angular deviation in the aberration acts over a longer distance. This makes the aberration pattern grow more in size relative to the airy disc which should be constant in both cases as the f-ratio is the same

Overall aberrations will be worse with this arrangement but optical modelling would have to be carried out to see how much worse and it may be that this increased level of aberrations is only apparent for objects at low altitudes where the amount of aberration becomes significant.