Eyepiece spreadsheet

[Ruud]

Here’s a new version of the Scope Calculator. It’s an excel workbook and uses no macros. A few example sheets are included. One has the information of Don Pensack’s buyers guide (over a thousand eyepieces. The included information is partial, to get the complete information go to Cloudy Nights’ eyepiece forum. It's in a sticky.)

You can make extra sheets by right-clicking on the name of a sheet (at the bottom).

The spreadsheet: ScopeCalculator v1.xls

Input (yellow cells):

• Telescope’s name, aperture, focal length, central obstruction (yes/no).
  
• Observer’s dark adapted pupil, preferred maximum and minimum exit pupils.
  
• Eyepieces names, focal length, apparent field of view, known field stop (if available).
  
• Magnification factors of  Barlows, reducers, field flattener, coma corrector and such.
  

Output:

• Calculated field stop for zero angular magnification distortion (zero AMD). When you don’t know the field stop diameter of the eyepiece, the calculated field stop determines the true field of view in the telescope. A bullet indicates that the eyepiece cannot fit in a 1.25" barrel.
  
• Rectilinear distortion (RD). This depends on the apparent fov of the eyepiece alone. It is always of the pincushion type.
  
• Angular magnification distortion. This is calculated from the focal length of the eyepiece and its known field stop. When the AMD is positive it causes extra pincushion distortion (on top of the RD). When the AMD is negative it causes barrel distortion. Zero AMD is best. When the AMD is very big and positive  (>10%) or very small and negative (<-5%) your known field stop may be wrong and a field stop warning is given by turning the eyepiece data red.
  
• Magnification. In the absence of AMD this is for the whole field of view, otherwise it is just for the centre.
  
• True field of view. When the field stop is known, the TFOV is corrected for the calculated AMD. The TFOV gets bigger when the AMD is negative and smaller when it is positive. When the field stop is not known the AMD is assumed to be zero.
  
• Exit pupil. Green cells are for wide field, yellow cells for deep sky (nebulae and galaxies) and red cells for planets. A red font indicates that the exit pupil is outside the observer preferences. When the telescope has a central obstruction and the exit pupli is larger than the observer’s dark adapted pupil the font is strike-through. It is also strike-through when the exit pupil is too small.
  

Options you can change

• Maximum field stop that fits a 1,25" barrel. This is set to 29.2 mm, but that is just a guess.
  
• Field stop warning (yes/no). If you set this to no, the colour coding of the cells in the AMD is switched off,  just as the red font when the known field stop is possibly wrong.
  
• Exit pupil formatting (yes/no). You can use this option to turn off the colour coding of the exit pupil column.
  

Please note

• When entering numerical values, just enter the numbers. Units like degrees or mm are added automatically.
  
• When you copy and paste data into the eyepiece columns, use the right click menu and choose “paste special  > values”. If you just use Ctrl+V or paste you’ll erase the conditional formatting of these columns.
  
• My excel version has a bug that will sometimes add bullets to eyepieces that aren’t 2 inch. Sorting the columns will make the excess bullets go away.
  
• With some of the columns, sorting can move the eyepieces to the bottom of the sheet. When this happens, use undo, select the rows with the eyepieces, and sort again. Or just go to the bottom of the sheet. The header rows remain visible when you scroll down.
  
• When you first sort by focal length from largest to smallest, and afer that by name from A to Z you get the eyepieces sorted by name and subsorted by focal length. This is the nicest way to view the Don Pensack data.
  
• Minimize the ribbon to see more rows.
  
• The formulas are written over more than one line. Make the formula window larger to see them properly.
  

A screenshot. The Televue 40mm Plössl has an unusual amount of AMD. Possibly, the field stop as reported by Televue is incorrect.

A small animation to illustrate AMD:

[Chris Hopson]

Wow, nice work mate.

Chris

[John]

Thats an interesting piece of work

Where the spreadsheet indicates optical performance characterstics such as AMD, RD etc, etc, are these figures the results of actual tests on the eyepieces themselves or are they derived in some other way ?

Thanks.

[Ruud]

Hi John, it's all maths and definitions. 

The maths (all angles need to be expressed in radians):

RD is calculated from the form the afov. It is independent of the quality of the eyepiece.

If theta is the angle of the ray that is farthest from the optical axis, then, by definition, RD = (tan(theta) - theta) / tan(theta). This distortion comes from projecting a sphere onto a plane. Two eyepieces with the same afov always have the same RD. RD is always of the pincushion type.

If it weren't for RD, he diameter of the field stop would be 2 * tan(theta) * (eyepiece focal length).

When you correct for RD the field stop becomes 2 * tan(theta) * (eyepiece focal length) * (1 - RD). Fortunately, this simplifies as Field Stop Diameter  = afov * EFL

Without AMD the field stop would be afov * EFL, but this is not always the case. When the known field stop differs from the calculated field stop, AMD is present.

AMD is defined as (afov after distortion - afov without distortion) / (afov after distortion). The afov after distortion is the manufacturer provided value of the afov. The afov without distortion is the known field stop divided by the eyepiece's focal length. When the distortion blows up the afov, the AMD is positive (pincushion) and adds to the pincushion distortion from the RD. When the distortion reduces the afov, the AMD is negative (barrel) and detracts from the pincushion distortion from the RD.

So, RD is always present and of the pincushion type, while AMD may be zero, negative (barrel) or positive (pincushion).

AMD has an influence on the quality of an eyepiece from the astronomer's perspective.

Positive AMD is bad. The afov is inflated and the true field of view is smaller than you might expect from the afov. TeleVue Wide Field and Takahashi Ultra Wide have a lot of positive AMD. 

Negative AMD gives a true field of view that is larger than you might expect from the afov. This is not preferred for astronomy though, as it will make the angular separation of double stars seem smaller at the edge of the field than in the centre of the field.

Negative AMD, however, reduces the pincushion of the RD, which is nice for terrestrial eyepieces (like the Docter), where straight lines like the horizon don't become as curved when seen at the edge of the field.

To calculate the AMD of an eyepiece you need to know the field stop. Most manufactures seem reluctant to provide it. TeleVue, Pentax, Baader and Docter are exceptions.

Field stops can be measured though, with a calliper when they are external (Plössl for instance), or by the drift time method. The latter uses the time that a star needs to cross the field of view in a stationary telescope. A star close to the equator is preferred, and it has to pass right through the middle of the eyepiece.

Here's a workbook that will help you with drift time field stop measurements, should you want to try it:

Drift time field stop workbook: FieldStop.xls

[John]

Thanks for that.

Out of interest, what would your reaction be if an owner of an eyepiece or a manufacturer said that the eyepiece concerned did not, in actual use, display the optical characteristics that your calculations indicate ?

[Ruud]

Rounding explains most discrepancies. The Ethos 13mm for instance could have any focal length from 12.5 to 13.499 mm. If you take the focal length to be 13mm precisely the AMD would be 1.7%, whereas Nagler says it nowhere exceeds 1%. If you take it to be 12.9mm the AMD would be 1%.

For the TV 40mm Plössl I suspect a mistake in TV's list. It is stated to have a 27mm field stop. Given the afov of 43° the AMD would be 10.1%. That is an awful lot compared to the other TV Plössls! At the same time, an 1.25" barrel does have more room than 27mm for the field stop. I wish they had an e-mail address to contact TV to ask them what's going on.

I'd like to know more about the Meade megawides. I've read that their afov is closer to 82 than to 100 degrees. I'm not sure how that has been determined. If it is from tfov x magnification, then AMD is ignored. A big chunk of AMD might bring their afov close to 100° while their true field is more like that of a run of the mill 82°.

All three Nagler T4s have very little AMD, by the way.

Anyway, in the spreadsheet I've chosen for field stop warnings when the AMD is below -5% or above 10%.

[YKSE]

Ruud,

I have a couple of questions, hopefully can get simple answers from you.

1. It seems to me that your numbers are from calculations, not from actual measurement, correct?

2. Where is source of definition of Rectilinear Distortion(RD) and Angular Magnification Distortion(AMD) you're using?

For my simple mind, rectilinear means straight lines, therefore Rectilinear Distortion(RD) means that straight lines are not straight any more, no matter the lines bending outwards like ) | ( (pincushion distortion) or inwards like ( | ) (barrel distortion);  while AMD means the angular distortion, the lines are kept straight throughout the FOV (unless AMD is coupled with RD which do happen in many eyepieces), while there's angular distortion in the FOV.

For the TV 40mm Plössl I suspect a mistake in TV's list. It is stated to have a 27mm field stop. Given the afov of 43° the AMD would be 10.1%. That is an awful lot compared to the other TV Plössls! At the same time, an 1.25" barrel does have more room than 27mm for the field stop. I wish they had an e-mail address to contact TV to ask them what's going on.

3. I think there're some 1.25" eyepieces have even 29mm field stop, e.g. Orion Ultrascopic or Baader eudiascopes. Is it possible that Televue knows what field stop their 40mm plossl has and your definition of RD/AMD calculation is wrong?

[RT65CB-SWL]

Thank you Ruud.  

Both files downloaded. I will have a 'play' with them when I have nothing better to do.   

[Ruud]

Hi YKSE,

1. It seems to me that your numbers are from calculations, not from actual measurement, correct?

Yes, it's all calculations, based on the values entered in the yellow cells.

2. Where is source of definition of Rectilinear Distortion(RD) and Angular Magnification Distortion(AMD) you're using?

Rutten & Venrooij Telescope Optics Evaluation and Design, Chris Lord and Telescope Optics.net.

RD    =   (tan(theta) - theta) / tan(theta)  =  1 - theta / tan(theta)

AMD =   ( theta₁ - theta₀ ) / theta₁                  =   1 - theta₀  / theta₁

For zero AMD, theta₁ (the angle from which the ray actually comes) equals theta₀ (the angle from which the ray should come). Theta is the angle of a ray with the optical axis. For the field stop calculation we are interested in rays that graze the field stop. For those rays theta = afov / 2 (in radians).

I would prefer different definitions for RD and AMD, of the type (new - old) / old. Obviously convention disagrees with me. EP designers prefer (new - old) / new.

For my simple mind, rectilinear means straight lines, therefore Rectilinear Distortion(RD) means that straight lines are not straight any more, no matter the lines bending outwards like ) | ( (pincushion distortion) or inwards like ( | ) (barrel distortion);  while AMD means the angular distortion, the lines are kept straight throughout the FOV (unless AMD is coupled with RD which do happen in many eyepieces), while there's angular distortion in the FOV.

Both AMD and RD can be of barrel or pincushion distortion type. The RD we're concerned with arises from projecting a spherical view (the afov)  on a flat plane (the focal plane at the field stop). RD is of the tan(theta) type.

Terrestrial eyepieces are preloaded with barrel-type AMD which is intended to cancel out the pincushion RD. In the animation above the cancelling out is perfect in the one extreme (terrestrial eyepiece) and completely absent in the other extreme (telescope eyepiece).

3. I think there're some 1.25" eyepieces have even 29mm field stop, e.g. Orion Ultrascopic or Baader eudiascopes. Is it possible that Televue knows what field stop their 40mm plossl has ... 

You are right. A 29mm field stop fits easly in a 1.25" barrel. Some manufacturers even squeeze in a 29.2 mm field stop. Yet, TV's field stop for their 40mm 43° Pössl, according to TV, is 27 mm (see here).

I think the 27mm field stop is a typo. 29mm makes a lot more sense. With a 29mm FS the eyepiece would have an AMD of 3.3% which is normal for a Plössl.

... and your definition of RD/AMD calculation is wrong?

I have three reasons to think that the answer to this one is no:

1) The calculated RDs are in perfect agreement with Lord. 

2) The calculated field stop for zero AMD is:

2 * tan(theta) * (eyepiece focal length) * (1 - RD)

In an earlier version I had a column with the calculated field stop including the AMD, using:

2 * tan(theta) * (eyepiece focal length) * (1 - RD) * (1 - AMD) 

That was a useless column because it had values that were exactly equal to the known field stop for each and every eyepiece. That cannot be a coincidence.

3) This means that if TV is right about the 27mm, the calculated RD of 6.4% (=0.064) agrees with Lord and together with the calculated AMD of 10.1% (=0.101) the calculated field stop agrees with the 27mm field stop.

And if TV made a typo and the 40mm Plössl actually has a field stop of 29 mm, the calculated RD and AMD of 6.4%=0.064 and 3.3%=0.033 are again in perfect agreement with Lord and the 29mm field stop.

The typo is more likely. It resolves the mystery of the unused 2 mm, and brings the AMD down to a normal value.

I hope that answers your questions, Yong. Thanks for being a critical reader and asking the questions.

[YKSE]

Hi YKSE,

1. It seems to me that your numbers are from calculations, not from actual measurement, correct?

Yes, it's all calculations, based on the values entered in the yellow cells.

 Thanks.

Hi YKSE,

2. Where is source of definition of Rectilinear Distortion(RD) and Angular Magnification Distortion(AMD) you're using?

Rutten & Venrooij Telescope Optics Evaluation and Design, Chris Lord and Telescope Optics.net.

RD    =   (tan(theta) - theta) / tan(theta)  =  1 - theta / tan(theta)

AMD =   ( theta₁ - theta₀ ) / theta₁                  =   1 - theta₀  / theta₁

For zero AMD, theta₁ (the angle from which the ray actually comes) equals theta₀ (the angle from which the ray should come). Theta is the angle of a ray with the optical axis. For the field stop calculation we are interested in rays that graze the field stop. For those rays theta = afov / 2 (in radians).

That's what I understand too.

I would prefer different definitions for RD and AMD, of the type (new - old) / old. Obviously convention disagrees with me. EP designers prefer (new - old) / new.

For my simple mind, rectilinear means straight lines, therefore Rectilinear Distortion(RD) means that straight lines are not straight any more, no matter the lines bending outwards like ) | ( (pincushion distortion) or inwards like ( | ) (barrel distortion);  while AMD means the angular distortion, the lines are kept straight throughout the FOV (unless AMD is coupled with RD which do happen in many eyepieces), while there's angular distortion in the FOV.

Both AMD and RD can be of barrel or pincushion distortion type. The RD we're concerned with arises from projecting a spherical view (the afov)  on a flat plane (the focal plane at the field stop). RD is of the tan(theta) type.

seems to me that you have strong opinion about what should RD and AMD

... and your definition of RD/AMD calculation is wrong?

I have three reasons to think that the answer to this one is no:

1) The calculated RDs are in perfect agreement with Lord. 

2) The calculated field stop for zero AMD is:

2 * tan(theta) * (eyepiece focal length) * (1 - RD)

In an earlier version I had a column with the calculated field stop including the AMD, using:

2 * tan(theta) * (eyepiece focal length) * (1 - RD) * (1 - AMD) 

That was a useless column because it had values that were exactly equal to the known field stop for each and every eyepiece. That cannot be a coincidence.

In Lord's paper, the calculation is done for total Geometrical Distortion (GD), from I learned from Mr. Lord, it incluldes both RD and AMD, and if you assume AMD is zero, of course you RD=GD, you'll get AMD=GD too if you assume RD is zero.

Can you provide the source of your formula for calculating field stop?

[Ruud]

Hi YKSE

You know, there are actually more than just RD and AMD. Think of a tilt-shift lens for cameras, Such lenses introduce anti-perspective distortions to set walls straight for when you can't take an architectural photo head-on. This is also a geometric distortion. But to get to the point:

Assuming that RD is zero is a bad idea. That would only be approximately true for eyepieces with a really narrow field of view. RD only depends on the afov, and two eyepieeces with the same afov have the same RD. As soon as the afov is known, the RD is known.

When the AMD is unknown you cannot include it in a calculation of the field stop. Ideally, for telescope eyepieces, AMD is zero. So if you assume AMD to be zero, you get an "ideal field stop".

Ideal Field Stop  =  2 * tan(theta) * (eyepiece focal length) * (1 - RD) 

The (1 - RD) part in this formula is the correction factor for rectilinear distortion. Lord's graph for geometric distortion E is actually a graph of RD. His description of E (RD) is a bit incomprehensible. He says it is "calculated by taking the ratio of theta and tan-theta and expressing it as a proportion of the latter". It turns out that RD is:

RD  =  (tan(theta) - theta) / tan(theta)  =  1 - theta / tan(theta)

It could have been defined differently, like blue over red, but, it is at it is, blue over yellow, and with this definition the correction for RD is "times (1 - RD)".

The 2 * tan(theta) * EFL part in the Ideal Field Stop formula is how you would calculate the field stop assuming no RD. It is the way you would calculate the base of a isosceles triangle with altitude EFL and vertex angle afov, where theta is half the afov. 

Corrected for RD, the ideal field stop becomes:

Ideal Field Stop  =  2 * theta * EFL = afov * EFL

From Lord, I only used the first graph for the geometrical distortion E (= RD) to check if I had my RD defined in the conventional way. Lord's graph definitely assumes that AMD=0. It really shows RD alone. He is just not very clear about it.

For the rest, the ideal field stop agrees with Rutten and Venrooij. They don't give the definition for RD, but say that the edge of the field stop would be located at y = f * tan (beta) without correcting for RD, and that it would be at y = f * beta after correcting for RD. 

I do not assume AMD to be zero. It just ideally is zero and a field stop for that condition is worth mentioning I thought.

The point of the workbook is to calculate a better true field, so I calculate the AMD whenever possible, from the known field stop. With "known" I mean the value provided by the manufacturer or a value measured with either a caliper or the drift time method. 

Unfortunately, measured values of field stops available on the internet are often wrong. For instance, for my 16mm Skywatcher Nirvana the reported field stop varies from 23 to 25mm. From my experience, I suspect that the Nirvana has a slightly negative AMD and that its true field stop is 23.3mm. Ideally, the field stop for this eyepiece would be 22.90mm.

Note that in the column for the "calculated field stop assuming AMD is zero" the font is light blue when a known field stop has been entered, and dark blue when the known field stop is missing. There's also colour coding in the column "true field of view": the font is green when the AMD could be calculated and dark blue when it could not be calculated. 

[YKSE]

Hi Ruud,

Hi YKSE

Assuming that RD is zero is a bad idea. That would only be approximately true for eyepieces with a really narrow field of view. RD only depends on the afov, and two eyepieeces with the same afov have the same RD. As soon as the afov is known, the RD is known.

How do you know it's an bad idea? In Chirs Lord's paper you put the link, there's this paragraph, Quote(I red mark the relevant sentence):

"Angular magnification distortion is corrected to a large extent in astronomical eyepieces, and rectilinear distortion in binocular & spotting 'scope eyepieces intended for terrestrial use. The point to bear in mind is, that if one is made constant with field radius, the other is left uncorrected and manifests itself to the extent shown in the geometric distortion plot. "

 Lord's graph for geometric distortion E is actually a graph of RD. His description of E (RD) is a bit incomprehensible. He says it is "calculated by taking the ratio of theta and tan-theta and expressing it as a proportion of the latter". It turns out that RD is:

RD  =  (tan(theta) - theta) / tan(theta)  =  1 - theta / tan(theta)

RD SGL.png

It could have been defined differently, 

I'm glad to know that you prefer different definitions about optical terms than Chris Lord has been using.

Ideal Field Stop  =  2 * tan(theta) * (eyepiece focal length) * (1 - RD) 

The (1 - RD) part in this formula is the correction factor for rectilinear distortion.

If I'm not totally mistaken, this calculation formular is deducted by you with your optical knowledge. I'd recommend you to contact Christ Lord, his mail address is in that link (I've done that about some questions about that paper and he has kindly sent me the source codes for his calculation in that paper), he may very well find your work very interesting.

[Ruud]

This is what Lord says:

"Angular magnification distortion is corrected to a large extent in astronomical eyepieces, and rectilinear distortion in binocular & spotting 'scope eyepieces intended for terrestrial use. The point to bear in mind is, that if one is made constant with field radius, the other is left uncorrected and manifests itself to the extent shown in the geometric distortion plot. "

This is what he might have said

"Angular magnification distortion is avoided in astronomical eyepieces, and negative AMD is used in terrestrial eyepieces to correct the pincushion caused by RD. The point is that without negative AMD you'd see the RD.

No wonder he confuses you.

RD is always there. For an 82° eyepiece it's always 17.7%, irrespective of how much AMD is present. You can't assume that it is zero. It's just like you would not assume gravity to be zero for an object suspended from a string. On Earth, for a 1 kg object, gravity is always 9.81 N. Yes, the object doesn't fall, but assuming that its gravity is zero would be a bad idea. Because it's wrong.

It really does not matter that I prefer a different definition for RD. Optical engineers have their conventions, many of which dating back to the time of slide rulers, and they have practical reasons for those conventions.

I feel we should stop discussing Lord. Maybe you would like to read Rutten and van Venrooij's book. They express themselves clearly and telescope design is a very interesting topic.

Clear skies.

[John]

Using that tone Ruud, I doubt that anyone will want to join in discussion with you

[Ruud]

Oh, I'm just getting desperate because I keep having to talk about Lord. Don't worry. It'll pass.

Apologies to anyone who feels offended, directly or vicariously.

[YKSE]

Ruud,

I do read some pages of Rutten and van Venrooij's book, as a matter of fact, here's a couple of pictures as proof:

I think the relevant page is here:

http://www.telescope-optics.net/eyepiece_aberration_2.htm

[Ruud]

Hi Yong,

Ah! that's the private discussion we had about my struggle to find the proper definition of RD. I remember it well. 

The link you give is not to Rutten and van Venrooij's book.

It's a link to telescope-optics.net, an interesting source, full of information. Unfortunately its definition of RD contains an error. It says that RD = 1 - tan(a') /tan(a') which is zero for all a', so I gathered it must be wrong and I could not use it.

The book I recommend is by Rutten and van Venrooij, Telescope Optics, Evaluation and Design, A comprehensive Manual for Amateur Astronomers, published by Willmam-Bell.

I have the 1988 edition. It looks like this one on Amazon.

The latest edition has a slightly shorter subtitle. It is now

Rutten and van Venrooij, Telescope Optics, A comprehensive Manual for Amateur Astronomers, published by Willmam-Bell.

The link is to Willmann-Bell's website, where you can find a description of the book.

[michael.h.f.wilkinson]

As I understand it from my training in optics as a part of my university studies is astronomy (Kapteyn Astronomical Institute, University of Groningen), a rectilinear distortion is defined as any deviation from projecting straight lines as straight lines (usually measured most easily in imaging systems). In the case of rotationally symmetric optics this means a radial distortion. The simplest two of these are pincushion and barrel, but more complex forms exist. It is well known that you cannot simultaneously correct for RD and AMD to a fair degree of accuracy over a wide field of view. Given this definition of RD, I do not see that the field of view determines the RD. Rather, the field of view determines what the minimum achievable total distortion will be ("sum of RD and AMD" if you will). Changing the definition of RD to some abstract concept does not help understanding the problem for the average person

[michael.h.f.wilkinson]

BTW, in imaging lenses, FOV and rectilinear distortion in the sense used above are definitely not directly related. A Carl Zeiss Distagon wide-angle lens design (retrofocal, for use in (D)SLRs) has quite a bit of pincushion distortion. A Carl Zeiss Biogon of the same focal length and FOV, which works on cameras without moving mirror between the rear element and film or sensor has far less rectilinear distortion. I therefore do not see how this could not happen in the case of EPs. Different designs show different degrees of rectilinear distortion. It makes no sense to me to say: given the FoV this lens must have a given rectilinear distortion, it is just that you cannot see it. If lines appear (nearly) perfectly straight there is (nearly) no rectilinear distortion, by the definition of the term.

[Ruud]

Hi Michael,

Concerning your first post: Yes, that's true. Just about any distortion will distort a straight line not going through the centre of an eyepiece.

The sort of RD I calculate is the kind that you get from viewing a flat field (the image at the focal plane inside the field stop) as a spherical afov. If you don't take it into account and simply use  distance from edge of field stop to the optical axis = focal length * tan (afov/2)  you get values for field stops that are too large. 

R&V in chapter 16.3 of Telescope Optics put it like this:

"For zero rectilinear distortion the following relationship should apply:

y = f * tan (beta)

where y is the off-axis distance in the focal plane, beta is the image angle from the optical axis, and f is the focal length of the eyepiece. For astronomical eyepieces, however, it is important that angular magnification remains constant ... in this case the following relationship should apply

y = f * beta

where beta is expressed in radians. With zero angular magnification distortion straight lines in a focal plane appear curved in a pincushion fashion, with the curvature becoming greater the farther they lie from the centre. It is impossible to correct an eyepiece for both".

Their beta is my theta. It is the angle that a ray, grazing the field stop, makes with the optical axis when it leaves the eyepiece. They don't say more than quoted here on the issue, but if you draw a cross section of the afov and the field stop (with the focal length as unit) you get some insight in what's going on:

Now, whether AMD is zero or not, the arc theta projects on the focal plane as tan(theta), not as theta itself. This is my reason for saying that there is always RD in an eyepiece, and you get more of it when the afov gets bigger.

When you define RD as RD = (tan(theta) - theta) / tan(theta) the correction factor for calculating the field stop is (1- RD) and the field stop diameter becomes

2 * tan(theta) * (eyepiece focal length) * (1 - RD)

This simplifies to Field Stop = focal length * afov in the absence of AMD. Like Rutten and Venrooij say it ought to be. (In their terminology y is half the field stop and beta is half the afov.)

I applied the reverse of the above reasoning to calculate the AMD. With the known field stop and the focal length of the eyepiece you can determine what the afov would be if there were no AMD. This leads to a theta₀ which differs from half the afov, theta₁. I discussed this above, where I say

AMD  =   ( theta₁ - theta₀ ) / theta₁  =  1 - theta₀  / theta₁

For zero AMD, theta₁ equals theta₀. (theta₁ is the angle from which the ray actually comes, theta₀ the angle from which the ray should come in the absence of AMD). Defined like this, the correction factor for a true field stop diameter including AMD is (1 - AMD) and the field stop can be calculated as 

Field Stop Diameter = 2 * tan(theta) * (eyepiece focal length) * (1 - RD) * (1 - AMD) 

or, simplified

Field Stop = focal length * afov * (1 - AMD) 

You could use this in the rare case that the AMD is given instead of the true field stop. When you already know the effective field stop this formula just confirms it.

Concerning your second post:

A camera is not an eyepiece. In a camera, the sensor is directly in the focal plane of the objective. A transformation from a flat focal plane to a spherical afov does not apply. RD can well be zero in a camera (or really close to it) when you photograph a rectangle straight on. In the focal plane of a camera only the distortions of the objective matter.

Still, for the eyepiece of a DSLR the same reasoning applies as for any other eyepiece.

(In special circumstances like when you photograph the inside of a globe from its centre with a non-distorting lens, the lens still produces a photograph which shows the surface of the globe distorted. I suppose neither of us are talking about this, though. Hence the tiny letters)

I knew the RD-AMD issue is a tricky one. That's why I made the animation

It shows how, in an eyepiece, plenty of AMD can correct all pincushion from RD. Even for a ridiculously wide eyepiece like in this example.

Note that the full RD becomes visible when the AMD is zero: the arcs are of equal length but the segments are not. Pincushion will be seen.

Also note that the RD can be corrected for with negative AMD. This makes the segments of equal length but the arcs become shorter toward the edge of the field. Globing will be seen, but Pincushion is corrected for. Still, even in this case, a spherical afov is projected on a flat plane. There is still RD, even though it is corrected for with AMD.

For an eyepiece, is impossible to have no pincushion and no AMD at the same time.

Eyepieces with a tiny field, however, can come close to this condition (orthoscopic eyepieces). This is because when RD is below 6% you just don't see it, and AMD is not needed to correct for it. Orthoscopic eyepieces usually have an afov of 45° or less, and an RD of 5.2% or less.

I try to meet two criteria with my approach.

There needs to be internal consistency. There is, because 2 * tan(theta) * (1 - RD) * (1- AMD) invariably returns the known field stop. Of course, you may say, because that is a tautology. But it needs to be a tautology, just like x² - 2x + 1 = 0  therefore x = 1 is a tautology.

There also need to be external consistency. I suppose I have RD right because correcting for it gives the field stops you would expect without AMD, as reported by Rutten and van Venrooij. Also, I have read (I forget where, I guess I never thought that I would be writing essays on the subject) that the RD for an 82° eyepiece is 17.7%. Furthermore, Lord put up a graph that repeats the RD values I calculate.

There is one thing I ignore in the workbook: the distortions of the objective. that is on purpose. I calculated how much RD would arise from my TV Genesis 500 mm 5.6° objective assuming no AMD. The Genesis has a short focal length objective for a telescope. The RD turns out to be 0.0079, so small that ignoring it seems justified. For longer focal length objectives RD is even smaller. Objectives are much more orthoscopic than any eyepiece existing. RD and AMD can both be sufficiently close to zero. therefore I assume them both to be zero.

In the case of camera lenses which are often much shorter than 500 mm the RD of the objective cannot be ignored, and neither can AMD, but that is the topic of your second post above concerning cameras. Photozone.de has some nice examples of camera lens distortions. 

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Now, Michael, how would you go about calculating the RD and AMD of for instance the TV 15mm Widefield? I have that eyepiece. Pincushion in it is HUGE. The eyepiece is in the sheet Experiments 2 of the workbook.

TeleVue says: fl = 15 mm, afov = 65° and the field stop diameter is 15.4 mm. I calculate RD = +11% and AMD = +9.5%, both are of the pincushion type and they add to each other. (Simply adding the two percentages is probably incorrect, though.)

Also, if you know my reasoning is invalid, could you please give some indication to the right way to approach eyepiece distortions? I'm not asking you to write a complete essay, but maybe you could use the 15 mm TV Widefield as an example and add a few short annotations to a more correct calculation, indicating your motivations. I'd be much obliged. We all would be!

Thank you, Michael.

And thanks everyone for their patience.

[michael.h.f.wilkinson]

As I understand it you are effectively looking at relative displacements of points along a line through the centre of the FOV. This is actually much harder to measure in practice, if the target is just a straight line. Furthermore, it is not rectilinear distortion in the accepted sense. Because your RD is not the standard RD then that just confuses the matter and you should use a different term. In the classical definition RD, AMD and FOV are interrelated or more correctly, the optimal values of AMD and RD are related to FOV, you can do worse than the optimum. You could say the best EPs lie along a Pareto optimum front in terms of AMD and RD. What you are effectively doing is eliminating one variable (RD), by introducing some new variable (say RD') expressing that as a function of another (FOV). Mathematically sound, provided you make quite clear that RD' ≠ RD, which was not the case through the use of the same word/symbol.

In terms of physics, or engineering, RD' is not that useful, exactly because it is a function of FOV. It doesn't add anything to the understanding of the properties of the EP.  Lets take the example of the distortion test I performed on several EPs. No real measurements, just a visual inspection on a linear target. I would be hard pressed to assess RD', but good old RD is easy: just look at how lines bend. Now take to EPs of similar FOV: the XW7 at 70 deg and the Delos 8 at 72. If I reduce the field stop of the Delos to get the same 70 deg I still see more RD (pincushion) than in the XW. The XW must have more AMD if we assume both EPs are at or close to the Pareto optimum front (which would be my guess).

Thus, I am not saying your maths are wrong, except that symbols are not used consistently with common practice.

[Ruud]

Thank you Michael,

I wish I knew what the common practice is with the symbols. But it's nice to know that the maths are right.

So the RD might better be named RD'. It is the pincushion inherent to an eyepiece with a given apparent field of view. You'd see it the the absence of any other distortion.

The spreadsheet does not intend to calculate the quality of an eyepiece. It just calculates what can be calculated from the afov, focal length and the known field stop. And the calculated values of RD' and AMD only apply to the edge of the afov.

For observers who want to measure angles outside the centre of the field, or compare angles across the field, it is merely a hopeful sign when the calculated field stop  is equal to the known field stop.

For observers who want an eyepiece that suppresses pincushion, it is a merely a hopeful sign if the calculated AMD is a few percent less than zero. (I'm such an observer, I like straight lines and don't care much about measuring angles.)

The resultant distortion for the view as a whole cannot be calculated from the entered values. For such a calculation one needs to know the glasses used, the curvatures of the lens surfaces and the thickness and spacing of the lenses. The ray tracing program Oslo can do the calculations, but you need all vital statistics of the eyepiece to start with.

The strange distortions can be occur. Kunming UO makes an inexpensive 70° line of which the 20mm shows the most peculiar distortion. The image bulges in the centre and looks as if projected on a blister. Outside the centre this distortion is absent. The transition between the blister and the rest of the field is quite  sudden which makes the blister very noticeable.

The spreadsheet can not calculate or predict such an effect. Compared to Oslo, Excel is limited and the ambitions of the spreadsheet are too. 

When you input values that are wrong, you get output that is useless. Especially when you find a value for a field stop diameter, check it with the manufacturer or with what others measure. You can also measure a field stop yourself with a calliper or, when it is internal, with the drift time method.

Good luck to all.

[michael.h.f.wilkinson]

I would not call it "native pincushion"; that is misleading. It is in fact the worst pincushion distortion along the Pareto front, when AMD is made zero. RD_max might be a better term, in fact

[Ruud]

Hi Michael,

I agree with you that it is a bad idea to use the term RD for the rectilinear distortion that results from the projection of the eyepiece’s spherical afov on the (hopefully) flat focal plane of the objective. As you rightfully say, calling it RD suggest that this projection is the only source of rectilinear distortion. It is not. So, I’ve moved on, and now call it RD’. 

There is a one-to-one relation between RD’ and the afov. When theta is half the afov, RD’ = 1 - theta / tan(theta). In a way, one could say that for eyepieces RD’ is inherent to the afov, as I did. It is a function of the afov. As you say above: "RD' is not that useful, exactly because it is a function of FOV".

To me, RD' is useful, as it allows me to calculate the field stop in the absence of AMD and other distortions. I find that field stop interesting. This because for some purposes, zero AMD is ideal. Comparing angles on the sky is just one of them.

So, I find that calling RD' by this name is a good idea, but I'm not sure it would be better to refer to it as RD_max . RD_max might suggests that the total rectilinear distortion cannot exceed the rectilinear distortion that results from zero AMD alone. After all, for many eyepieces AMD is not zero. AMD can have a negative as well as positive non-zero value, introducing barrel or pincushion distortion, decreasing or increasing the pincushion that results from RD'.

That the total RD can be greater than RD' is demonstrated by the TeleVue WideField eyepieces. 

You've probably heard the term King of Pincushion used for the Panoptic. If it deserves that nickname, its predecessor, the TV WideField, deserves the title Emperor of Pincushion. I have one. The afov is 65° and the fl is 15 mm. The field stop is 15.4 mm. RD' for this eyepiece is 11%. If there were no AMD the field stop would be 17.02 mm. In reality it is 15.4 mm. The difference is big and AMD at the field stop is +9.5%. You have to look through it to believe the pincushion it has. Much more than you would expect from a 65° eyepiece.

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I'm exploring a new idea.

It's a tentative idea, but this is it: The resultant distortion from RD' and AMD (in the absence of any other distortion) might be calculated as

      RD _{from RD' and AMD}  =  1 - (1 - RD') * (1 - AMD).

• For the TV WideField that yields  1 - (1 - 0.11) * (1 - 0.095) = 0.195. That is 19.5%. A lot of pincushion.
• For the 14 mm Pentax XW it gives   1 - (1 - 0.128) * (1 + 0.029) = 0.102. That is 10.2%. Less pincushion than the WideField, yet the view is wider.
• In case of the 14 mm Morpheus you get  1 - (1 - 0.151) * (1 + 1.015) = 0.138. Or 13.8%.

Here's a v2 beta of the scope calculator that contains a column using this approach. One sheet contains example data. There's another sheet that's empty so that you can try it for a telescope and eyepieces that you own.

The  v2 beta workbook>>>   ScopeCalculator v2 BETA.xls

There is only one difference with ScopeCalculator v1: a column has been added for the combined calculated distortions RD' and AMD. Again, no other distortions are assumed. AMD is calculated at the field stop. The workbook does not aim to calculate the quality of the eyepieces. It cannot do that. The data in the populated sheet are manufacturer supplied.

Bye