Sweet spot calculation, mathematics of eyepiece optics

[kimmik (Kim)]

Curiosity got to me today, and I spent the last hour figuring out how to calculate the size of an eyepiece's sweet spot.

http://upload.wikimedia.org/wikipedi...Bicone.svg.png

Turned out rather interesting - the shape of the sweetspot is a biconical volume like the picture below, and depends on a few factors. If anyone knows references, can you link it in?

Sv - sweet spot vertical length (tip to tip distance of the bicone)
Sh - sweet spot diameter (diameter of the fattest cross section)
Svol - sweet spot volume
Pe - exit pupil diameter of the scope/eyepiece system
P - eye pupil diameter
AFOV - apparent field of view

Assumptions:
SAEP - spherical aberration of exit pupil is negligible
The boundary of the sweetspot is defined as the surface on which, if you placed the axis of your pupil there, you still see the entire AFOV, with vignetting no greater than this figure of overlapping circles - one circle is the exit pupil, one circle is the eye pupil.

http://cdn-7.analyzemath.com/Geometr...problems_1.gif

If P > Pe, then:

Sv = P/Tan(AFOV/2)
Sh = P
Svol = πSvSh^2/12
Svol ∝ P^3

So, I measured my pupil to be 6mm fully dilated, which gives me these results:

Sh = 6mm
AFOV 50° - Sv 13mm Svol 0.12mL
AFOV 68° - Sv 9mm Svol 0.085mL
AFOV 82° - Sv 7mm Svol 0.066mL
AFOV 100° - Sv 5mm Svol 0.047mL

If your eye pupil was say, 8mm, then Sh and Sv proportional increase by x1.33, and Svol increase by x2.37

If you were in daylight, and eye pupil was 2mm, assuming P > Pe, then Svol is now x0.037 or 3.7% of before. That explains why eye positioning is much more critical during the day, unless the exit pupil is big, in which case you use Pe instead of P for the above calculations. Pe=6mm will give the Svol as P=6mm.

Eye relief distance of the eyepiece doesn't affect the above values unless its very short, in which case the vertical sweetspot begins inside of the eyepiece.

Spherical aberration of the exit pupil will reduce all of the above values, Sh Sv and Svol. I have not yet figured out how to incorporate it into the calculation, maybe someone else could give it a go?

[kimmik (Kim)]

Worked out the effect of SAEP for a specific example, my 20mm 100° explore scientific eyepiece, with rated 15mm eye relief. But I wasn't able to simplify the maths into something presentable.

If there was no SAEP, and eye pupil is 6mm:
Eye position is 12.5-17.5mm vertically, and +/-3mm horizontally.

With SAEP and eye pupil 6mm:
Eye position is 11-16mm vertically (effective eye relief is 13.5mm), and +/-2.5mm horizontally.

Now if you try to use this eyepiece in daytime with eye pupil of 2mm, assuming the exit pupil is also 2mm:
Eye position is 13-14.5mm vertically, and +/-0.5mm horizontally! Outside this cubic millimeter of space you will get see partial blackout and kidneybeaning.

In extension to the first post, here's a diagram that shows how the sweet spot of eye placement works.

[Wavytone]

Kim,

You're only looking at eye relief and spherical aberration of the exit pupil (the "kidney bean" effect). This is by no means the most important - and you've omitted the most important part - the telescope. Not all eyepieces suit all telescopes - quite the contrary as most have found out the hard way. What you see are the sum of the aberrations of the telescope AND the eyepiece. It is possible for example to have an eyepiece that suits a refractor that is very ill-suited to a fast Newtonian, and vice versa. Similarly an eyepiece that suits a Meade or Celestron SCT probably won't be great in a refractor.

In short:

a) it is essential to know the focal ratio of the telescope you will use this with, and the corresponding depth of field; what works well at f/15 or f/10 probably won't work at f/4.

b) you need to compute - by- raytracing - the field curvature, astigmatism and coma of the telescope,

c) find an eyepiece design that works well with an incoming beam of the same focal ratio as the telescope across the whole of the field of vie, i.e. at the edge of the field stop as well as the centre of the field. Many eyepieces work well at f/10 but are rather poor at f/4.

d) choose an eyepiece that has field curvature that is approximately the same as that of the telescope. If the field curvature is at least of the same sign its promising; if it is of opposite sign the eyepiece + telescope combination will be terrible.

e) find an eyepiece design that has coma and astigmatism opposite to that of the telescope, the net effect being to cancel the contribution from the telescope.

f) eyepiece designers usually don't bother to correct distortion and put that in the too-hard basket.

Beyond that.. if you are really into custom optics it is possible to match the chromatic aberrations of a refractor and eyepiece. This is not done for consumer grade eyepieces, however some Zeiss refractors were designed this way, where it was assumed the scope would always be used with the particular eyepiece (i.e. the eyepiece is not interchangeable). Some military equipment - e.g tank and gunsights - are also designed this way, and many people buy them on the surplus market only to find they are not suited to use with commercial eyepieces.

From memory Conrady provided the formulae for designing a complete Dialyte telescope (fast singlet objective, smaller negative singlet lens halfway down the barrel and a matching two-element eyepiece) that performed rather well for its day, but the eyepiece and telescope only worked as a matched pair and could not be interchanged with anything else.

By way of example it is fairly obvious that Televue and Explore Scientific are designing their eyepieces to suit fast Newtonians around 30-50cm aperture at f/4, and have built-in some degree of negative coma and matching field curvature. There is simply no way that any eyepiece could give pinpoint images across a 40mm focal plane in an f/4 telescope without this. These same eyepieces will perform adequately in an f/10 schmidt cassegrain (Meade or Celestron) because the depth of field is enough to accommodate some of the field curvature, however in a fast refractor (f/7) they perform rather poorly because the field curvature doesn't match and the depth of field is too shallow to cover this.

For the same reason the humble Plossl eyepiece turns out to be an excellent match with Newtonians as it (surprise surprise) also has a curved focal plane that is a close-enough match and negative coma.

[kimmik (Kim)]

Thanks for the comprehensive explanation, indeed that was my intent, to deal with the effect of SAEP and the eye pupil positioning. I do understand that there are many other aberrations.

As far as i know, field curvature, astigmatism and coma are field aberrations, and have relatively minor effect on SAEP. You dont hear people complaining about XW or Delos having SAEP in one scope but not in another.

Regarding matching those field aberrations though, other than trial and error, there also seems to be minimal information available, and it would be good to compile a list of optic properties.

[Wavytone]

Kim,

I don't see why you are so hung up about spherical aberration - in most eyepieces this is not an issue. It usually becomes a problem in big low-power eyepieces where the exit pupil is comparable to the observers eye pupil, and for this to be a real problem, this depends on the focal ratio of the telescope you are using. An eyepiece that has intolerable SA on one telescope (eg a fast Newtonian) may well be perfectly ok on an f/10 or f/15 scope.

Over the years there have been a few notable (terrible) exceptions - one specific Nagler was well known for it.

Secondly you can't really calculate spherical aberration of the exit pupil of an eyepiece like that.

Most designers use ray tracing to evaluate eyepiece performance, then calculate the field curvature, coma, lateral chromatic error and deal with these first. SA is looked at as a check, usually as the last priority.

One valid way to calculate it is to ray trace the entire eyepiece an for that you need to know the radii, thicknesses and the glass types for each element in the eyepiece. This data is however proprietary to each manufacturer, ie basically there is no way you can acquire this data.

The only other valid way is to measure the aberrations of an actual eyepiece on an optical bench.

[kimmik (Kim)]

i think you have it wrong way round. small exit pupil exacerbates the blackout/kidney beaning, not big exit pupil.

and how do you know i didn't ray trace the SAEP, and measure my 20mm 100deg....