Exit Pupil & Eyebox: Very Crude and Preliminary Hypothesis (Refinement needed)

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Minuteman
Jan 30, 2020
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Disclaimer: I am not an optical engineer and am not remotely an expert on this subject. I simply waste too much time thinking about it. If you are more informed on the topic, feel free to provide your own insight on the mechanics of eyebox. As it stands, I think there are too many inconsistencies for my speculation to really be quite correct, but it might be a starting point.

In addition, this conjecture revolves primary around low power scopes. I'm not certain if higher power optics could be affected differently by this proposed dynamic.


Most of this is simply copied and pasted from ARFCOM.

Perhaps the most commonly referenced stat in relation to eyebox size is exit pupil. This is basically a point where the rays of the "cone of light" emitting from the optic come together at the technical eye relief point. Here's an illustration of it:

1920px-Exit_Pupil_raytrace.svg.png


The width of that disc is the exit pupil diameter. As the popular conception goes, this circle must overlap with the pupil of your own eye in order to see an image. Larger exit pupil = Larger eye box, and more area to move your head side to side without losing FoV completely.

The problem with this is that it does not seem to align very well with practice. While optics with large exit pupils do not usually have outright poor eyeboxes, optics with small exit pupils occasionally have very large eye boxes, and 2 optics with similar exit pupils may have different levels of forgiveness. In addition, if one shines a light through an optic and measures the circle for themselves, they will find that it becomes smaller if you are closer than the listed eye relief, yet the eyebox becomes larger. (The FoV starts shrinking eventually, but that's not germane to the point.) How can this be?

I just chalked this up to reality being more complex than paper, but I've come up with something that I think may provide an actual mathematical basis for this.

Recently I remeasured the eye relief (distance from ocular to eye) on the Elcan SpecterDR 1/4x. Previously I thought the listed number of 70mm was exactly correct; actually the ideal eye relief distance seems to be about 55mm. At 70mm you begin to encounter scope shadow and the eyebox decreases in size. However, the number of 70mm is correct for the calculated exit pupil (32/4=8mm). At 55mm of eye relief, the disc of light is actually only about 6mm wide on 1x.

This suggests to me that the technical eye relief and exit pupil distance is actually not necessarily the point where the scope has its most favorable optical performance. In many cases, it is likely closer than stated.

In addition, my attempted measurement of the SpecterDR's eyebox size on 1x is about 18mm. In order for this to correspond with the exit pupil, my own eyes' pupils would have to be about 5mm wide while looking out of a sunny window on a TX afternoon. This would mean I probably have some of the largest pupils in the world (average in these kind of brightness conditions is about 2-3mm), to the point where I probably wouldn't even need NVGs to see in the dark, and yet I don't seem to see significantly if any better in the dark than the average person.

So here's where the whole eyebox calculation thing comes into play: My assumption is that the actual eyebox corresponds with the cone of light illustrated above. If you are closer than the stated eye relief, it is larger than the exit pupil; if you are further away, it is also larger. This is because it shrinks to its minimum point where the light rays all come together. To find the size of this cone, you need to know 1) how wide it is to begin with, 2) your distance from the optic, 3) where it becomes smallest (technical eye relief) and how wide it is there (exit pupil). I will further assume that the size of this cone increases and decreases proportionally with distance.

In that case, with the Elcan's 1x magnification, the size of the cone starts at what I will assume is the objective diameter (32mm), and it becomes smallest at 70mm away, where it shrinks to 8mm. Getting closer than this will increase cone width proportionally, depending on distance, up to 32mm.

55/70 = 78%, and the number at the 78% waypoint between 32mm and 8mm is 13.15mm. This seems to line up almost exactly with the recorded eyebox (~18mm) and a normal eye pupil size of 2.5mm.

This would also explain most of my experiences and impressions of the eyeboxes of different optics.

Elcan vs TA02/TA31 4x ACOG: Both have the same exit pupil (8mm) yet the ACOG has a larger side-to-side eyebox. It is not drastically larger but the difference was enough to notice the first time. The ACOG also has an objective diameter of 32mm. With a minimum eye relief of about 1.1 inch, this would suggest an eyebox cone of about 14.4mm. (1.1/1.5=0.73, the 73% progression point from 32mm to 8mm is 14.4mm.)

Primary Arms GL2x vs Trijicon 1-6x Accupoint/Credo: Both have 24mm objectives and exit pupils of about 12mm, but the GL2x seems to be much more forgiving of side-to-side movement. It also allows for a lot of back and forth head positioning before FoV shrinks. Assuming minimum eye relief were, say, 2" on the GL2x, but was the stated eye relief on the Trijicon, the GL2x would have an eyebox cone of about 17.15mm (2/3.5=0.57, 0.43*12+12=17.15) versus 12mm on the Trijicon.

TA44 1.5x ACOG: The calculated exit pupil of this optic is large but not remarkably so (10.67mm), smaller than some LPVOs at 1x, yet it seems to very easy to get behind from reports. Perhaps this is largely because the maximum eye relief is actually very long, as compared to the listed eye relief of 2.4" (which I assume nobody actually shoots with on a TA44). Assuming a distance of 4.5", the eyebox cone would be about 15.44mm.

TA33 ACOG: The listed exit pupil is merely 8.4mm, yet again this scope seems to have a very large eyebox. With a postulated eye relief distance of 3.5", knowing the objective is 30mm and the listed eye relief is 1.9", we could calculate a diameter of 26.58mm. (3.5/1.9 is 1.84, assuming the eyebox cone increases linearly with distance past the listed eye relief, 26.58mm is 84% of the way to 30mm from 8.4mm.)

Elcan vs Swarovski Z6i: The Elcan's eyebox seems larger despite having a smaller exit pupil (9.6mm vs 8mm). If, say, the Swarovski's ideal eye relief distance were 3.1" away, versus the listed distance of 3.74", then with a starting cone of 24mm it would have an eyebox cone of about 12.06mm versus the 13.15mm calculated for the Elcan earlier.

Variable optics in general: Often times drawing back from the minimum eye relief will cause a decrease in overall eyebox size, where the FoV will 'blink out' rapidly if you move your head significantly off to the side. However, you can move your head more without encountering any decrease in your observed FoV. Perhaps this is where the calculated exit pupil is encountered, and maintaining 100% of your FoV while moving your head requires your eye's pupil to fall completely within the disc of light. Since the overlapping disc is larger but the overall cone of light is smaller, you see the phenomenon where you can move more without getting any scope shadow, but you cannot move as much before blacking out completely.

Upon drawing your head further back yet, the eyebox increases, because the cone of light starts dispersing outwards and increases in size. (Oddly enough, the disc of light also increases in size - I really do not know how that works.) It is the likely reason why NX8 users report they can get better results on 1x by mounting their scope further forwards.

This would also explain why optics with small exit pupils sometimes have big eyeboxes but optics with large exit pupils never really seem to have small eyeboxes, from what I've heard. The eyebox cone will never be smaller than the exit pupil, so with a large exit pupil optic, you will be getting at least that amount regardless of optimal eye relief distance. With a small exit pupil, you may or may not have a small eyebox depending on cone starting diameter and distance from technical eye relief.

Possible questions and areas for future inquiry:

Q: How can you get a full FoV when the light rays aren't intersecting?

A: This is something I honestly have no idea about, and is where my lack of optics engineering knowledge becomes apparent. All I can do is report that some optics do maintain full FoV, or even only get their full FoV, at distances closer than where they achieve their calculated exit pupil.

Q: If eyebox diameter were also influenced by pupil size, then why doesn't the eyebox increase in size in low light conditions where my pupils grow? If anything, eyebox seems to become worse in dimmer conditions.
A: This is one of the major difficulties encountered when trying to reconcile pupil size and eyebox. A possible explanation is that this is because, in low light conditions, the outer edges of this cone are simply not bright enough for your eye to perceive the transmissions as a coherent image. This seems consistent with how any additional scope shadow in dimmer lighting seems to be more of a soft haze than a hard black edge.

Q: If eyebox diameter were equal to light cone + 2x pupil diameter, then how would we explain the 10x on a 1-10, or the 18x on a 3-18x, where the eyebox seems to be significantly smaller than 7-8mm, or even 5mm? According to this model, the eyebox diameter would have to be at least 5mm wide with an eye pupil size of 2.5mm, no matter what.
A: Zero clue, and a very interesting point to consider. Perhaps it's the case that the light intercepted at the edges of our eyes' pupils is also less useful, and that the nature of the cone on high magnifications exacerbates this somehow? though that seems rather far fetched. Perhaps higher magnifications mimic the case of low light mentioned above, where the edges of the cone are not bright enough, but this happens even in broad daylight? If anyone has a technical explanation for this, feel free to chime in. The eyebox = scope light + eye pupil thing seems to be the most doubtful portion of this whole thing, but this would require an alternative explanation as to how eyeboxes can also be considerably larger than the estimated light cone diameter.

Q: The number for the TA33 seems too large. In addition, it seems the TA33 ACOG also has a very large eyebox up to its minimum eye relief distance around 1.9", even though the exit pupil is small (8.4mm).
A: Yet another wrench, and one that would require more looking into. It's optics like these that really make the whole prospect questionable.

Q: Would the eyebox cone's starting diameter actually be equivalent to objective diameter, or would it better match the size of some other lens nearer the eye?
A: This is also something that I simply don't know. It could also make sense that the cone diameter would actually match up better with the ocular (eyepiece) lens diameter, or the lens right before it; however this doesn't seem to work very well for the GL2x, TA33, and TA44 since they have smaller oculars than many LPVOs. (The 4x ACOG's is not particularly large, either.)

Q: What is the minimum/maximum eye relief you're referring to?
A: I use this term to describe the furthest point at where you can maintain maximum FoV within the scope. Moving further forward/back results in decreased FoV or scope shadow. Usually full FoV eye relief characterized by hard, defined outlines (field stop, presumably) surrounding the scope picture (as opposed to the "softer" fuzzy image edges when outside this range) and, unfortunately, more scope housing obstruction.

Full FoV eye relief:
SHOT-2020-9.jpg


21320c9df94956b2251f3af760953953.jpg

2nd Picture Credit: Primary Arms

Note hard borders around edges.

Outside of full FoV range:
33e651319e2261ab4173326f8348c330.jpg

(Also from Primary Arms)

Notice soft, slightly fuzzy edges.

Usually the minimum or maximum eye relief is also where you find the largest eyebox, so it is also often the most optimal point to be at, as long as you don't mind the scope housing.

Q: What do you mean the TA31 has a large eyebox, the eyebox sucks
A: Note that for these purposes I am defining eyebox as how much side-to-side play you can get while still maintaining some sliver of the FoV. I am not talking about the front-to-back axis.

However the ACOG's front-to-back forgiveness is not bad either IME, it's just that the eye relief distance for full FoV is very short. This can be compensated by moving the optic back on your rail to suit your natural head position so that you see the hard edges described above. Once the optic is setup this way my experience is that it is very easy to get behind, potential scope eye issues notwithstanding.
 
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There are quite a few folks on this venue that are qualified to give you a definitive answer on this.

In an earlier time, the classical term "Ramsden disc" was used to describe what is now called the exit pupil of an optic. Incidently an optic usually has both an entry and exit pupil. This exit pupil appears as the image of the front objective as it hits the rear of the optic and has a certain diameter in width.

The term "Ramsen disc" is from Jesse Ramsden, an English instrument maker.

The Ramsden disc/exit pupil is the image of the front objective (usually reduced in size) which is hitting the rear eyepiece of an optic. This disc contains whatever you can see through the front objective, which at first, appears very small, and appears to be "dancing around in a sea of black inside the eyepiece", which becomes progresively larger as you bring the eyepiece closer to your eye and when you've reached the correct distance/eye relief, and when the optical centerline axis of your eyeball and the optical centerline axis of the optic merge as one straight line, you then see the image coming through the front objective as it fills the eyepiece.


The term "eyebox" as folks are using it, is how they perceive how the image they see at the rear/eyepiece of an optic as that image appears to be either fairly "steady" or to "dance" around given that you are at the correct distance from the eyepiece (eye relief).


When the centerline optical axis of your eye and the centerline optical axis of the optic merge as one straight line at the correct distance (eye relief), you'll see the image coming through the front objective as you're supposed to see it.


The trouble is, and/or when this image you're seeing begins to "dance around" which is because of the movement of either your eyeballs or the optic in relation to each other where the merged centerline axis of both the optic and your eyeball, momentarily break into separate lines.

The movement of your head/eyeballs in relation to being lined up behind the optic is causing the image to "dance around" as you look at it through the eyepiece.


The appearance/perception of an "eyebox" is an illusion, you don't look inside a scope, what's inside the scope or lens, projects out of the optic and hits your eyeball at the correct distance (eye relief).



The most extreme example of this is the old movie projector which throws an image against a wall maybe 40 ft away.

You want to verify this, look at the schematic the OP included in this discussion.


Maybe it's me misunderstanding what I've read but FOV doesn't have anything to do w/the idea of an eyebox, so whatever/however wide/narrow FOV there is, is an issue of what's coming through the front objective (assuming the coverage of the eyepiece in relation to the exit pupil/what's coming thru the front objective).


Also look at the first image which is a pic taken behind a scope. You can look at it and see that the cell phone/camera, whatever took the image, isn't lined up correctly behind the scope.

The "giveaway" is the fact that the "dark ring" at the rear of the scope isn't even in thickness because one side is closer to the optic of whatever took the image than the opposite side of the ring causing that side of the ring to be distorted in size (the foreshortening effect), so that the rear of the scope isn't plano-parallel to the front of whatever optic took the image (and/or not centered one behind the other), simply, it's "cockeyed".

Thus, the axis of one optic won't be lined up w/the axis of the other optic, they're 2 separate lines instead having merged into one.

The rear of the scope and the optic which took the image behind the scope are mis-aligned where the centerline axis of both optics aren't lined up along a common axis, and even though you can see an image, it isn't going to reflect the best performance of the scope.

The resulting image might still look good, but it's "cockeyed", and it's wrong. As I've mentioned somewhere else, if you get a scope shipped to you, and the rear element is "cockeyed"/not lined up correctly behide the other lens elements, not being crazy, you'll send it back.

That's because the optic w/the rear element "cockeyed"/will not perform the way it's supposed to, and it's the same EXACT problem/issue if you have a separate optic/optical system, taking a pic behind the scope that's also "cockeyed", which doesn't work either for the same reason.


The issue of the rear of the scope "looking fuzzy" is simply that the optic behind the scope is focused at distance, anything close to the lens of the optic which is behind the scope is going to appear out of focus. Simple as that.



A final thought. The exit pupil of an optic has a certain diameter which means something in relation to the pupils of your eyeballs (also a change in magnification as it affects the exit pupil of the optic). As you age, the pupils of your eyeballs will shrink to where when U reach about 70, they're at about 2.5mm.

The width of the exit pupil of an optic in relation to the size of the pupil of your eyeball will affect just how light or dark you perceive an image to be that's coming through a scope.

So 2 different folks looking through the same scope can have a different perception of how bright or dark what's coming through the optic appears to be.

Because of your individual eyesight, the same scope that appears bright to U can appear dark to somebody else.

That's my take on this.
 
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Let me just say this too. The folks that make these "top tier" optics, are doing so w/incredible precision. Many of the outfits making these scopes aren't "Godzilla" like Carl Zeiss, they're small "boutique" outfits, and I doubt they're making the kind of money off these scopes people think they are, and there's no guarantee they'll survive, let alone the vendors that supply them.

I think one needs to be careful/take what's spread around the internet w/a grain of salt, some of the comments/criticisms bandied about these optics that don't give them a "fair shake", a chance.

Because of someone's individual eyesight (cataracts/astimatism,/color blindness/"old eyes" et al), a bright and very sharp optic can "look like shit" and "darker than what I expected" where these comments that show up on a venue are more about the eyesight of the folks looking through the optic than what the optic is actually capable of, but in spite of that, the optic gets a bad rap.

Pics can lie too, so I'll emphasize that if at all possible, and sometimes I know it isn't possible, you try to look to get a look through an optic firsthand.

I knew my eyesight was getting bad because of how tough it was getting to use my camera gear.

I was not going to buy/spend close to 4 grand on the March HM until after I had my implant surgeries and healed up and verified w/my surgeon that my vision had improved to where I could take advantage of the detail and nuance that optic was/is capable of.
 
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If there's any confusion about the centerline optical axis of an optic, or the idea that line merges w/the centerline optical axis of another optic, then look at the schematic the OP included @ the start of this discussion.

There is only ONE STRAIGHT LINE which goes through the center of the optic and the exit pupil which is represented by the line in the center of the schematic. This line is the centerline optical axis of the optic.

The same line/straight line would represent the centerline optical axis of your eyeball if someone uploaded a schematic of an eyeball.

When you're lined up behind the scope @ the correct eye relief, this line in the schematic of the optic, and the same line/the centerline optical axis of your eyeball will be the 2 lines which merge into one line, presenting the image as you should see it, and when there's movement between your head/eyeball and the back of the scope, they will break off into 2 lines when the image starts to "dance around".

The idea also counts if you have a cell phone/camera behind a scope where the optical axis of the 2nd optical system behind the scope isn't lined up with the optical axis of the scope which means the pics you take under these circumstances won't reflect the true performance of the scope.







1666303357534.png
 
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For the sake of clarity re The issue of taking pics behind a scope w/a cell phone and getting an idea of just how well they're lined up behind the scope this is what I'm talking about.

The thicker side of the ring at the back of a scope in a pic means that side is distorted because it's CLOSER to the cellphone than the other side and means the optic of the cell phone taking the pic is at an angle the to the rear of the scope so that the optical axis of the cell phone won't/cannot be lined up w/the optical axis of the scope.



Ring.jpg
 
Appreciate your thoughtful and thorough responses Convex, that hopefully will help many understand the limitations and misrepresentations of through the scope images which is why I stopped trying to do that years ago and instead try to explain to others to not use through the scope images as a judge of optical quality, but more for an idea of reticle and/or FOV at various magnifications (but even here there can be deviations from what the eye sees when properly aligned).
 
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As i know eye box isn't just eye relief, it's also includes the exit pupil measurements allowing you to not have a perfect cheek weld. Eye relief is great but if the exit pupil is small, it's still going to have a small eye box.
 
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I asked a major scope manufacturer tech support person the question, Does a larger exit pupil provide a better eye box?

Here is their response:

Yes, there is a correlation between exit pupil and eye box. The larger the exit pupil, the more forgiving the eye box will be.