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:
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:
2nd Picture Credit: Primary Arms
Note hard borders around edges.
Outside of full FoV range:
(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.
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:
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:
2nd Picture Credit: Primary Arms
Note hard borders around edges.
Outside of full FoV range:
(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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