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Eyepiece AFOV calculation: figures don't seem to add up

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#1 popovkos

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Posted 26 May 2015 - 08:59 AM

Hi,

I was doing an exercise lately calculating an eyepiece AFOV directly from its geometry. It seems that this angle can be calculated as AFOV = 2 arctan(d / 2f), where d is the field stop diameter and f is the eyepiece focal length. Please refer to the attached picture.

Now, taking specs for, e.g., a 32 mm 1.25" Plossl from Televue website, d = 27 mm and f = 32 mm, we get AFOV = 45.7 degrees, good 8.6 % less than the claimed 50 degrees. Similarly, for their 40 mm Plossl, (d = 27 mm and f = 40 mm), the formula gives AFOV = 37.3 degrees, rather than the claimed 43 degrees.

What did I get wrong?

#2 csrlice12

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Posted 26 May 2015 - 09:38 AM

I think its the blue ink....achros don't do blue very well....

I would think the shapes and curvature of the lenses would also have to be considered.

#3 Rich V.

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Posted 26 May 2015 - 10:47 AM

I don't think there's anything wrong with your math and the formula you are using is accurate as well.

You have to take into account that claimed eyepiece AFOVs are generally figured by the "simple" method which only multiplies the TFOV x magnification.  It does not take into account the angular magnification distortion as you look towards the edge of the FOV.  For instance, if you go to the Televue eyepiece calculator page, you'll see that if you use the simple calculation of the resultant TFOV and mag. stated in their calculator, the 32mm Plossl works out to have a 48° AFOV.  Throw in a couple additional degrees caused by designed-in pincushion distortion and you have your 50° claimed AFOV.

You'll also see that Panoptics which are claimed to be 68° AFOV work out by simple calculation to be in the 63-65° range.  Their 24 Pan's AFOV by the ISO calculation is 58.56° but figured by simple mag. x TFOV it's 64.24°.  You still have to take into account the distortion characteristics of the eyepiece design as well; frequently a bit of pincushion distortion is designed in so that we don't see so much of the "rolling ball" effect that a distortionless eyepiece design causes when panning.  The Panoptics, for example, are well known for having some pincushion distortion.  Add a few additional degrees of AFOV for this and it's easy to see how they get their 68° number.

Those of us who use fixed power binoculars see this "simple" method applied all the time; a typical 10x50 bino with a 6.5° FOV will be claimed to have a 65° AFOV (10 x 6.5°).  Using the ISO 14132-1:2002 standard the calculated AFOV would only be 59.18°.

2ω’ = 2 x tan-1 (Γ x tan ω)
= 2 x tan-1 (10 x tan 3.25°)
= 59.18°

Rich

#4 popovkos

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Posted 26 May 2015 - 11:06 AM

Thanks Rich V., it makes sense.

#5 Jon Isaacs

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Posted 26 May 2015 - 11:30 AM

AFoV can be measured directly.  David Knisely has developed some techniques for direct measurement that are quite interesting.. Hopefully he will post a detailed description but basically one projects a beam through the eyepiece against a flat wall and measures the distance from the exit pupil to the wall and the projected diameter and use that to calculate the AFoV, it's quite amazing just how much bigger the circles are for the various fields of view..

Jon

#6 ChristianG

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Posted 26 May 2015 - 01:42 PM

Hi.

Personnally, I think the 'both eyes open' method is the easiest: Look at a wall with markings with one eye and look at the field stop through the eyepiece with the other eye. Simple trigonometry involving distance from the wall and width of wall that matches the field stop diameter will yield the AFOV directly.

By the way, the eye itself is adding a certain amount of barrel distortion when field of view approaches 70 degrees, so most eyepieces balance that with a certain amount of pincushion distortion. For more info, look into 'globe effect':

http://www.holgermer...distortion.html

In the image below, pincushion distortion is evident. But maximize its size on your computer screen and bring your nose closer while fixating the center, and at some magic distance, the distortion will disappear! Try it!

--Christian

#7 Rich V.

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Posted 26 May 2015 - 02:34 PM

For anyone interested, since calculating AFOV is an imprecise task depending on some unknown factors, as Jon points out, actually measuring AFOV is the only way to be reasonably accurate.

One way was brought up in the Binoculars forum a few years ago by Glenn LeDrew.  His method measures the angle of the light cone being projected out of the eyepiece against a wall.

Apparent FOV?  A quick way to measure!

Rich

#8 David Knisely

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Posted 26 May 2015 - 02:41 PM

Hi,

I was doing an exercise lately calculating an eyepiece AFOV directly from its geometry. It seems that this angle can be calculated as AFOV = 2 arctan(d / 2f), where d is the field stop diameter and f is the eyepiece focal length. Please refer to the attached picture.

Now, taking specs for, e.g., a 32 mm 1.25" Plossl from Televue website, d = 27 mm and f = 32 mm, we get AFOV = 45.7 degrees, good 8.6 % less than the claimed 50 degrees. Similarly, for their 40 mm Plossl, (d = 27 mm and f = 40 mm), the formula gives AFOV = 37.3 degrees, rather than the claimed 43 degrees.

What did I get wrong?

Well, I'm afraid that the thing that may have been wrong is attempting to calculate an accurate value for the apparent field of view of an eyepiece.  The formulae used are often just approximations (particularly the old AFOV = Mag*TFOV which is often 10% or more off) or just plain wildly off of reality, so to expect them to be accurate may be a little unrealistic.  The Apparent Field of View of an eyepiece is a measurable quantity only, and despite efforts to calculate it, it is what it is.  Here is how you can measure it:

..........MEASURING THE APPROXIMATE............
....APPARENT FIELD OF VIEW OF AN EYEPIECE....

"Both-eyes Open" Technique:

MATERIALS: 1. A Meterstick, Yardstick, or other linear device whose length is accurately known, which can be hung vertically on a wall, and whose exact middle or center is accurately marked. This could also be a narrow strip of paper of known length with its exact middle and ends marked clearly. This object will be known as the observing "target".
2. A method of holding and properly supporting an eyepiece rigidly in a horzontal position (like a bracket attached to a camera tripod), but which can be manually moved towards or away from a measuring target.
3. A tape measure.

STEP #1: Mount the vertical "target" (ie: the Yardstick or its substitute) on the wall so that its exact middle is will be about same height above the floor as the center of the eyepiece. For a meter stick, the midpoint will be the 50cm mark, and for a yardstick, it will be the 18 inch mark. Mark this midpoint with a visible marking like a small piece of tape or a black felt tip marker, so the middle can be easily seen from a distance.

STEP #2: Mount the eyepiece at a height above the floor which is exactly the same as the mid-point of the target, so that the observer can look into the eye lens with the eyepiece optic axis or barrel horizontal and parallel to the floor. Make certain the eyepiece is as horizontal as possible, and that it can be easily moved towards or away from a nearby wall from as little as two feet from the wall to as much as six feet away.

STEP #3: place the eyepiece straight out from the wall from where the observing "target" is located. Look into the eyepiece with *both* eyes open and merge the images of the eyepiece field of view and the target. Make the center of the superimposed eyepiece field centered on the mid-point mark of the observing target as closely as possible, and keep your head level with the floor (ie: keep your eyes at the same height above the floor).

STEP #4: Look at the top and bottom of the target, again with both eyes open. With both eyes open, try to make the top and bottom edges of the eyepiece field match the top and bottom edges of the target on the wall by carefully moving the eyepiece towards or away from the wall. Make certain when moving the eyepiece that it remains pointed exactly towards the center of the observing target, and that its height above the floor does not change. Once the edges of the eyepiece field match the top and bottom of the target, take the tape measure and measure the distance from the back of the eyepiece just beyond the eye lens (ie: where your eye was sitting when you were looking through the eyepiece) to the middle of the target on the wall. If the target has a length of "2Y" and the distance to the wall you measured is "D", then the apparent field of view of the eyepiece is then AFOV = 2*ATAN (Y/D), where Y is *half* the total length of the target and ATAN the arc-tangent (or inverse tangent) function. For example, if you were using a yardstick (36 inches in length, or Y = 18.0 inches) and your eyepiece field matched its length at a distance of 37.0 inches from the center of the target, the apparent field of view of the eyepiece would be about 51.8 degrees. You may have to look around a bit and be careful about eye placement to see the edges of the field stop and get the edges to line up properly with the outer target marks, so this method is not quite as accurate as the next one:

"Projection" technique:

MATERIALS: 1. Small flashlight with adustable front lens (Maglight is ideal).
2. A method of holding and properly supporting an eyepiece rigidly in a horizontal position (like a bracket attached to a camera tripod), but whichcan be manually moved towards or away from a measuring target. Also needed is a way to hold a flashlight horizontally so its beam goes directly into the field lens of the eyepiece.
3. A tape measure.
4. A wall with a large sheet of paper on it (butcher paper or newsprint is fine).

STEP #1: Mount the eyepiece so that the eye lens is facing the wall with the long axis perpedicular to the wall. Place the flashlight several feet away from the eyepiece and oriented so that its collimated beam can enter the field lens of the eyepiece.

STEP #2: Turn on the flashlight and turn out the room lights. Adjust the lens of the flashlight to produce a concentrated and roughly collimated parallel beam as it heads into the eyepiece.

STEP #3: Look at the wall and move the flashlight laterally until a large white disk appears on it (with wide-field eyepieces, only a portion of this disk may be visible at any one orientation of the flashlight). Measure the diameter of this disk.

STEP #4: Find the point behind the eye lens where the light from the flashlight appears to be coming to the most point-like focus. This is the "focal point" of the eyepiece. Then, measure the distance from that point to the middle of the projected light disk on the wall. If the projected disk has a diameter of "2Y" and the distance to the wall from the focal point is "D", then the apparent field of view of the eyepiece is then AFOV = 2*ATAN (Y/D), where Y is *radius* of the disk and ATAN the arc-tangent (or inverse tangent) function. Also, if you measure the distance from the focal point to the surface of the eye lens, that is the eye relief of the eyepiece.  This method seems to give more consistent values for the apparent field, but also in some cases will produce an apparent field of view that is slightly smaller than that obtained by the "both-eyes open" technique.

Clear skies to you.

Attached Thumbnails

Edited by David Knisely, 26 May 2015 - 03:34 PM.

#9 David Knisely

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Posted 26 May 2015 - 03:23 PM

I don't think there's anything wrong with your math and the formula you are using is accurate as well.

You have to take into account that claimed eyepiece AFOVs are generally figured by the "simple" method which only multiplies the TFOV x magnification.  It does not take into account the angular magnification distortion as you look towards the edge of the FOV.  For instance, if you go to the Televue eyepiece calculator page, you'll see that if you use the simple calculation of the resultant TFOV and mag. stated in their calculator, the 32mm Plossl works out to have a 48° AFOV.  Throw in a couple additional degrees caused by designed-in pincushion distortion and you have your 50° claimed AFOV.

You'll also see that Panoptics which are claimed to be 68° AFOV work out by simple calculation to be in the 63-65° range.  Their 24 Pan's AFOV by the ISO calculation is 58.56° but figured by simple mag. x TFOV it's 64.24°.  You still have to take into account the distortion characteristics of the eyepiece design as well; frequently a bit of pincushion distortion is designed in so that we don't see so much of the "rolling ball" effect that a distortionless eyepiece design causes when panning.  The Panoptics, for example, are well known for having some pincushion distortion.  Add a few additional degrees of AFOV for this and it's easy to see how they get their 68° number.

Those of us who use fixed power binoculars see this "simple" method applied all the time; a typical 10x50 bino with a 6.5° FOV will be claimed to have a 65° AFOV (10 x 6.5°).  Using the ISO 14132-1:2002 standard the calculated AFOV would only be 59.18°.

2ω’ = 2 x tan-1 (Γ x tan ω)
= 2 x tan-1 (10 x tan 3.25°)
= 59.18°

Rich

Actually, Tele Vue doesn't "get" the Panoptics 68 degree figure somehow: that is what it is.  I measured my 24mm Panoptic at 68.0 degrees +/- 0.2 degrees using the relatively easy techniques available to do that, and I expect Tele Vue measured it as well.  The Apparent Field of View is basically just a measureable quantity.  It is what it is despite efforts to calculate it.

The formula you cited needs to have its quantities defined more explicitly:

AFOV = 2*tan-1 (M*tan (TFOV)), where AFOV is the apparent field of view, M is the magnification, * is the mulitplication operation, and TFOV is the true field of view.  Let's try a case.  I have a 10 inch f/5.6 Newtonian with a measured focal length of 1410mm.  My old 30mm Orion Ultrascopic (26.08mm field stop) has a measured apparent field of view of 52.2 degrees and in the scope, it yields about 47x and a measured true field of view of 1.063 degrees on the sky.  Using the above formula, the AFOV would come out to be 82.2 degrees, which is way way off.  Even the old AFOV = Mag*TFOV yields a figure of 50 degrees, which is about 4.2 percent off of reality, but better than these tangent-based figures.

I recall the manufacturer of the old 5-8mm Speers Waler eyepiece making all sorts of wild claims about the eyepiece's apparent field (alledged to be variable from 81 to 89 degrees) based on using various formulae, yet measurement clearly showed it to be about 77 degrees and constant, no matter where the focal length of the eyepiece was set to.  Again, if you don't trust the manufacturer's figures for the apparent field of view of an eyepiece, you can measure it for yourself.  Clear skies to you.

Edited by David Knisely, 26 May 2015 - 03:54 PM.

#10 GlennLeDrew

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Posted 27 May 2015 - 12:13 AM

I've gotten the distinct impression over the years that some folks mistakenly think the AFoV is a kind of 'artificial' figure arrived at in approximation, by calculation using assumed distortion. As at least David, Jon and I stress, it's a very real, precise and directly measureable quantity. With this figure in hand, and in conjunction with the magnification and TFoV, the distortion--or at least its integrated extent at the field edge--can be calculated.

If any eyepiece manufacturer ever published an AFoV *NOT* based on actual measurement, or at least that figure which any suitably capable optical design software can so easily and accurately spit out, they would have to be ignorant dilettantes. Or given to misrepresentation (the more likely scenario.)

Edited by GlennLeDrew, 28 May 2015 - 04:55 AM.

#11 Starman1

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Posted 28 May 2015 - 12:04 AM

There is a way to calculate the maximum apparent field an eyepiece can have.

Measure the width of the eye lens.

Measure the eye relief unless you know it.

AFOV (max) =2 (Tan^-1[(0.5 diameter of eye lens)/(Eye relief)])

This does not tell you the actual AFOV, merely the maximum it could be.

And the eye relief figure to use is not from the lens unless the lens is flat--it is from a straight line from one side of the lens to the other.

You would subtract or add the sagitta of the lens to the eye relief depending on its concavity or convex nature.

This actually works and I've done it on a dozen eyepieces and come out very close to the manufacturer's claim.

However, I prefer David's method.  It's just hard to set up.

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