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Anyone knowledgable about Zoom Transition?

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Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/26/2008 at 11:48
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I'm creating an on-line low-light performance calculator for scopes that includes consideration for your eye pupil diameter as well as the perception of light intensity gains. It will provide graphical comparisons for multiple socpes. It's pretty much complete but I have a final question: Most variable power scopes that have a large objective lens and a low minimum zoom only utilize a portion of the objective lens diameter at minimum zoom. For instance the Zeiss Diavari Victory 3-12x56 has an objective lens diameter of 56 mm at 12x but only 44 mm at 3x. The 2.5x10x50 uses 50 mm at 10x but only 37.7 at 2.5x, while the higher magnification 6-24x56 uses the full 56 mm at both zooms. My question is, do scopes with smaller utilized minimal zoom objectives have a linear lens diameter transition all the way to maximum zoom, OR is there a mid-point at which the lens diameter reaches full diameter before maximum zoom?
Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/26/2008 at 13:36
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I wish you the best of luck with your project. It will be intersting to see.

I think there might be lots of problems to overcome. As one example, the Federal Glossary of Telecommunication Terms (FS-1037C) has the following: "(B)rightness should ... be used only for non-quantitative references to physiological sensations and perceptions of light." So, you can't use brightness as an interval or ratio variable to define low light.

For another, while it is easy to calculate the size of the apparent diameter of the objective lens from the size of your pupil and use that as a first approximation of the performance of the scope, other things can mitigate its actual and perceived performance.

Now to your question: The virtual image is formed using all of the light from the entire objective lens, irrespective of its size. But, because of vignetting, i.e. low light levels at the periphery of the image, stops can be used to limit the maximum size of the exit pupil. In my experience, as regards riflescopes, the maximum size of the exit pupil, even when limited, is larger than the maximum diameter of your eye pupil.
Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/26/2008 at 14:14
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With a formula suitable for the application, apparent brightness can absolutely be calculated from any reference point. This has been known for hundreds of years and the calculations have been completely refined beginning from the dawn of television to today's computer monitors. I've researched all of the evolved laws regarding this and I'm using the most applicable law in the most applicable manner. I will provide full details with the calculation.

Yes exit pupil/eye pupil relation is included in the calculation.

My question is regarding variable power scopes that do not utilize the full objective at minimum magnification.  I need to know that as the zoom increases from minimum, if the utilized objective diameter reaches full objective diameter at maximum zoom, or at some point in between. For instance, a 3-13x56 that only utilizes 44 mm at 3x, does it finally utilize the full 56 mm objective at 12x or does it utilize it some point like at 6x? Now that I think about it, since zoom only changes the objective-lens/exit-pupil relation linearly from one point to another, I just answered my own question.

Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/26/2008 at 15:04
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Originally posted by opticsmike opticsmike wrote:

My question is regarding variable power scopes that do not utilize the full objective at minimum magnification. 


You seem mathematically proficient, so think of it the relationship as a step function.
Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/26/2008 at 19:05
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The utilized objective diameter does not undergo a smooth transition from low magnification to high magnification.  There is a smooth transition between the lowest magnification and some intermediary magnification value where the whole objective lens is utilized.  That intermediary value differs from scope to scope, and some scopes utilize the whole objective lens at all magnifications.

Just out of curiosity, how exactly do you define apparent brightness and how do you plan to calculate it?

ILya
Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/26/2008 at 20:58
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koshkin, that was exactly the information I needed. I believe that the intermediary value can be deduced as long as the end points are known. It seems that only scopes with higher magnification to objective lens ratios are the ones the utilize the full objective at minimum zoom. This too should be able to be deduced for when manufacturers do not list the minimum utilized objective.

Apparent brightness becomes a highly involved discussion because it involves several factors. For a long time, it was generally accepted that the calculation was just one simple logarithmic formula very similar to the perception of sound based on the Weber-Fechner law, but then people learned that there's many more factors. A primary factor is what vision system is functioning, be it Photopic, Mesopic, or Scotopic, depending on the light level. Another primary factor is the amount of contrast involved. Another factor is in what FOV the object covers for each vision system. Eventually, formulas became exponential rather than logarithmic and there's argument that even the perception of audio should be exponential because the logarithmic calculation for sound induces a calculable error across wide differences in intensity. With vision, it's become more easily apparent that the formula is exponential and modern calculations for computer monitor light intensity scaling is exponential to display a brightness scale that is perceived linearly.

For the perception of brightness, the first objective is to determine which vision system is operating. Although Scotopic vision is more low-light sensitive than Photopic vision, Scotopic vision is very limited for scope use because of the lack of receptors within the 5+ degree center of vision which essentially creates a dark blurry view through a scope. Therefore, a primary necessity of a low-light optic is to elevate Scotopic light levels to the Photopic region, or at least to the Mesopic region which is a combination of Photopic and Scotopic vision. Therefore, I targeted the bottom end of the Mesopic region, as the central acuity of Photopic vision just begins to come into play to make the scope image usable for the brain to reconstruct. The formula is constructed by raising the light intensity difference factor to a power that correlates with the factors involved. If the difference of light intensity of the target through the scope is a factor of 50 times the intensity without the scope, the formula would be 50 to some power. With the factors involved, I found that a power between .25 and 1 was suitable, in which a judgement in overall conditions had to be made to determine the exact figure in between. In the end I settled on .33 because testing had been performed for the apparent brightness of a 5 degree target in the dark which provided the exponent of .33. So with that, a scope that delivers 50 times the light of the target to the pupil is perceived as 3.6 times as bright. That's with one eye so if you're comparing the scope on a single eye with both eyes open without a scope, the difference would be a factor of 25, resulting in a perceived brightness factor of 2.9, but for our intended purposes, we would still make the comparison with just one eye.

For anyone that is interested in learning more, there's a wealth of knowledge available. Do a search for Helmholtz, Gilchrist, Stevens' Power Law.

Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/27/2008 at 00:14
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Huh?  You brainiacs make my life difficult.  I am going to have to read this 50 times before I can understand it!  Whacko Big%20Smile
Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/27/2008 at 01:19
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Interesting.

A couple of comments:

from what I have seen that intermediary point where the scope is using the whole objective lens varies pretty wildly for different optical designs.  It is quite possible that it has to be empirically determined for each scope.

also, are you assuming a uniformly illuminated Lambertian light source that fills the field of view? if so, then how do you take into account a) black and white contrast of spatially structured targets? b) color contrast?

I did some experimentation on this a little while back with a calibrated light box and found that the relationship of the lab results to the field results can be quite tenuous.

ILya


Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/27/2008 at 04:38
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That's funny helo18, just trying to make it easier actually.

Not a uniform light source. And we're actually more concerned with reflectance unless we wish to hunt luminous aliens. Basically, perception of light intensity becomes increasingly proportional to actual light intensity as contrasts and light intensity decrease. Think of it this way, when the view is dull and dark with little contrast, we are able to more closely distinguish intensity variance. As brightness and contrast increase, we step up to perceiving increasingly larger intervals of light as if on a virtual linear scale, probably to eliminate the majority of redundant signal processing when enough information is already available to support the purpose of vision, just as all of our perception functions operate; even time by the way, that's why the years seem shorter as we get older, and the perception of time can also be calculated. Anyways, I am assuming a baseline reference as the point at which intensity and contrast are reduced to the point of difficult vision. It's the point between just under a full moon overhead on a clear night. From that level as a baseline and up, Photopic vision becomes involved and allows one to begin to be able to see through a scope, depending on the low-light performance of the scope. The final calculation is easy to adjust and I could provide a selection of environments to choose from.

It very well may be that there is no way to calculate the intermediary point of full objective utilization, but without some solution this project would be in vain. I noticed that scopes with higher magnification ranges in proportion to their objective diameters tend to have closer to full lens utilization at minimum zoom. I understand that an optical design may change things, but there still may be some fundamental underlying relation, possibly on the basis of that observation. I've tested 4 of my scopes this evening and that intermediary point is nearly in the middle of the zoom range for the most part. When I say in the middle, I mean in the middle of the zoom travel, not the middle of the numerical magnification range, as zoom does not change linearly. In the middle of the zoom travel, the actual numerical zoom is less than half. For instance on my 2.5-10x50, the middle of the zoom travel is at 5x, not 6.25x. If in the end there is no way to determine actual intermediary points without individual measurement, I may have to assume the middle of the zoom range as the safest average of possibilities. When running the possibilities through the program, a variance in intermediary point position in the zoom range by 25% has relatively little effect in relation to overall performance, as the most significant points are the begin and end points, and they are definitive.

If people wanted to get involved, we could have a database of measured scopes with pre-filled values so that all one would have to do is select a scope. Also, if anyone wants to get involved in the project development, send me a message. Most of the difficult work is complete and it just could use some fine tuning, specifically in regard to magnification transition. Once it is complete, it will be a great resource. At the very least it will be infinitely more useful than some twilight performance calculation, and it will be graphically presented with unlimited scopes overlayed for comparison. I'll likely limit it to ten scopes per calculation to conserve server resources.
Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/27/2008 at 09:10
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Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/27/2008 at 09:46
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Originally posted by Steelbenz Steelbenz wrote:

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No kidding!  I reread it again this morning and it makes more sense when I am not tired.  Maybe by the time they are done with this discussion, I can apply for a masters in light theory.

Thanks guys, for sharing you knowledge!  I learn a lot from it even if it does take me a long time to figure it out.
Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/27/2008 at 20:16
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just a question or two-- what transform are you using to get from the mostly linear physical system to mostly non-linear biological system?  have you included the frequency response or are you taking another approach?
Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/28/2008 at 02:40
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Transform? Frequency response? I'm not generating all of the background math and testing for this, I've researched what has been studied over the past few decades, most notably the past 10 years, and simply utilized the most applicable formula for target acquisition in low light. Ramen noodle soup is a great crack food for late night reading by the way. The only primary factors are the magnitude of light intensity gain, the vision system that is operating based on the light level, the field of vision (narrow), and the initial light intensity and contrast of the target. Believe me, I'm no guru by any means. I just did a ton of reading. Although I have done a bit of testing to see how accurate I feel it really is before and after I saw what the calculations would indicate what I should perceive.

Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/28/2008 at 03:25
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I'm beginning to think that a system of collaborative effort would provide the ultimate way to assist people in making their choices. If there is any interest, I may develop a freely available database of measured scope properties that can be compared with other scopes; properties such as resolution, contrast, light transmission, and stray light etc. Those whom have access to various scopes and wish to do a couple tests on them could become registered contributers to the database. These properties, in addition to zoom, objective, and exit pupil properties could provide a comprehensive analysis and comparison system.

Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/30/2008 at 07:58
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Sorry for the delay in responding to this thread, but I was out of town.

Because I have been looking for a riflescope for myself, my interest in a Z6 had taken me to the Swarovski website where I found that they list two values for objective diameter for their zoom scopes. The values appear to be the product of the objective lens diameter and the magnification. Perhaps this kind of information gives rise to the use of the phrase "utilized objective diameter." My thoughts about this follow.

The image we see has many characteristics, but I will address brightness, resolution and contrast in a very limited way.

By brightness I mean the total amount of light delivered to our eye, whether as a uniform or a modulated field. I was taught that the eye pupil is considered an extension of the exit pupil of the scope. In my experience, that is true. When the diameter of our pupil is smaller than the exit pupil, to a first approximation, we percieve the light as having come from an "apparent aperture" whose diameter is equal to the product of the magnification and the diameter of our pupil.

Example: 3-9x,40 set to 5x; eye pupil at 5mm. We will receive light as if it came from a 25mm lens.

Resolution: When light enters the circular objective lens, diffraction will occur at the outer edge. This unavoidable fact will cause every point in the image, even those on the optical axis, to be surrounded by a circle of light that is sometimes referred to as a circle of confusion. The size of this ring of light can limit our ability to resolve two close points (see my Avatar). As the true diameter of the objective increases, the circle of light gets smaller. Other factors are involved in resolution.

Contrast: The light due to diffraction from the outer edge of the objective lens, in addition to light scattered by anomalies throughout the lenses, light scatterd off the scope tube, etc., limits contrast.

Perhaps the above will explain my intital comment, "The virtual image is formed using all of the light from the entire objective lens, irrespective of its size."

Because of my personal interest in a riflescope, I had created a spreadsheet to help in my decision. Yesterday evening, I modified the spreadsheet to try to shed some light on the question about utilized objective. I think it underscores Koshkin's comments. There is too much info for me to type into this space, but I will send the file to one of the moderators.



Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/30/2008 at 09:46
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Optics Mike best of luck and keep us posted.
Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: September/30/2008 at 17:31
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Gunshow75, the relation you describe when you discuss brightness is absolutely correct. It is easily verified in my calculations and graphs. But the correct terms are luminous intensity and luminous flux, as brightness refers to our relative perception of these quantities. Having a smaller utilized lens objective at minimum magnification has no effect whatsoever on light gain. For instance, a 3-12x56mm scope that only utilizes 44mm at minimum zoom has the exact same exit pupil luminous intensity. Although the total exit pupil luminous flux is reduced, the luminous flux to the eye is not reduced because the eye pupil diameter is still smaller than the exit pupil diameter. For a 3-12x56mm scope at 3x... if it utilizes full the objective lens, the exit pupil luminous intensity is 9 times that of at the objective lens and the exit pupil diameter is 18.666mm. If the scope instead only utilizes 44mm objective at 3x, the luminous intensity is still exactly the same at 9 times that of at the objective lens and the exit pupil diameter is 14.666mm. The only effect of the smaller utilized objective at 3x is a smaller exit pupil diameter. Since the eye pupil diameter is still smaller than the exit pupil diameter and the luminous intensity is the same, the eye receives the same gain in luminous flux; for a 7mm eye pupil that would be 9 times the amount of light than without the scope.

On the other hand, if the full objective is not utilized by the time the magnification reduces exit pupil diameter to the size of the eye pupil diameter, the smaller utilized objective will produce less light within the eye pupil. For a 3-12x56mm scope, full objective diameter must be reached by 8x magnification for a person with an eye pupil diameter of 7mm, because the exit pupil diameter of 3x56mm is exactly 7mm. The luminous intensity at 8x is 64 times that of at the objective lens and transferred directly to the eye and perceived as roughly 4 times as bright as without the scope. Of course that is before considering the scope's light transmission.

Beyond 8x, the luminous intensity continues to increase by the square of the magnification, but illuminance is also proportionately reduced by the diameter of the exit pupil so that the same luminous flux is received within the eye at 8x magnification and beyond. So for someone with a 7mm eye pupil diameter, the only gain beyond 8x magnification is in target resolution, but no further light gain. As exit pupil continues to be reduced with increased magnification beyond 8x, although luminous intensity is increased for equivalent luminous flux at the eye pupil, the eye pupil begins to become less receptive and light gain can become reduced. Light gain can also be reduced with higher magnifications because scopes at higher magnification produce more stray light. Also, the circle of confusion reduces light gain by a factor of 0.84 because 16% of the light is forms the circle of confusion and 84% forms the image. The circle of confusion seems to only have an effect at smaller exit pupil diameters and so I'm having difficulting including this into my calculations. Does anyone know when that 14% should be included in the calculation for reduction of light gain? I'm sure that as zoom increases and exit pupil recreases that it's not an instant factor but rather has some form of increase to 14% loss.

I presume that scopes have smaller utilized objectives at minimum zoom so that the eye does not begin to experience diffractions with such proportionately larger exit pupils then the eye pupil. Or maybe it's to reduce the assembly length of the scope. I'm not sure.

Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: October/01/2008 at 07:03
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Originally posted by opticsmike opticsmike wrote:

Gunshow75, the relation you describe when you discuss brightness is absolutely correct. It is easily verified in my calculations and graphs. But the correct terms are luminous intensity and luminous flux,...


I am quite familiar with luminous intensity and luminous flux. For those who aren't, luminous flux is a quantitative expression of the brilliance of a source of electromagnetic energy in the range of 390nm to 770nm. It is measured in terms of the power emission per unit solid angle from an isotropic point source radiating equally in all directions in three-dimensional space.

Luminous flux is measured in lumens (SI units), which is 1 candela-steradian or 1.46 milliwatts at 550nm.

Luminous flux is a useful measure of the power emitted by a light source, but it is not used to compare brightness, which is a subjective measure.
Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: October/01/2008 at 12:27
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Luminous Flux at the eye pupil is the measure of the total amount of light delivered to the eye, where brightness is how we perceive its magnitude in relation to other amounts.
 
I just realized something that works out perfectly! Since scopes with smaller utilized objectives at minimum magnification must include the point of full objective utilization before the exit pupil diameter is reduced to the size of the eye pupil diameter by design, having a smaller utilized objective has no effect on the final calculation whatsoever and there is no need for obtaining the point of full objective utilization. I will still include the smaller objective diameter input at minimum magnification so that the exit pupil under low magnifications can be shown, and for those I will designate the full objective utilization point at 1/3 of the zoom range which is the typical physical midpoint of the zoom range, nearly where the scopes I've measured locate it. Any actual variance from that point will have no affect on the final calculation.
Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: October/01/2008 at 13:55
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Originally posted by opticsmike opticsmike wrote:

Luminous Flux at the eye pupil is the measure of the total amount of light delivered to the eye, ...

Luminous Flux is not a measure of the total amount of light delivered to the eye.   It is the instantaneous rate at which energy is radiated (dQ/dt) from a source of visible light.

In my opinion, the expression "utilized objective" is potentially misleading and not meaningful in discussions about brightness.
Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: October/01/2008 at 15:29
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The calculations are time independent. I guess I should have said that Luminous Flux at the eye pupil is the measure of the total amount of light delivered to the eye per unit time. Just as luminous flux is the amount of light emitted per unit time, it is also the amount of light received per unit time. The amount of light received from a source in not nearly equal to the amount emitted from a source, as the amount emitted is spread up to a full sphere and is then diminished by the square of the distance. The Troland is the actual term to describe the amount of light received to alleviate this ambiguity. I could very well instead utilize the term Troland, but most people are not as familiar with it as they are with the lumen.

The utilized objective only has an affect on the total received lumens (or Trolands) if the resultant exit pupil diameter is less than the eye pupil diameter.

Post Options Post Options   Thanks (0) Thanks(0)     Back to Top Direct Link To This Post Posted: October/01/2008 at 15:46
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Wouldn't that depend on the type of the source? or are you assuming a point source?  I would probably lean toward a lambertian emitter as a more appropriate approximation.

ILya
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Whether the source is evenly spread or a point, say the average illuminance from the source on a 56mm objective lens is one millilux (0.001 lumen/mm^2), then the amount of light per unit time emitted upon the objective lens is the illuminance times the area:

0.001 lumen/mm^2 x (3.14(56mm/2)^2)
0.001 lumen/mm^2 x 2461.76mm^2
2.46176 lumen

Now lets magnify that target by 7x with that 56mm objective lens:
Exit Pupil Diameter = Objective Lens Diameter / Magnification
Exit Pupil Diameter = 56 mm / 7
Exit Pupil Diameter = 8 mm

If that same amount of light is condensed into that 8mm exit pupil that corresponds with 7x magnification from a 56mm objective lens, then the average exit pupil illuminance will be:

2.46176 lumen / (3.14(8mm/2)^2)
2.46176 lumen / 50.24 mm^2
0.049 lumen/mm^2 (49 millilux)

If the original illuminance upon the objective lens is one millilux and the illuminance from the exit pupil upon the eye pupil is 49 millilux, then there is a 49x illuminance gain. This corresponds with a 7x magnification because the magnification is the virtual distance, or more precisely the virtual proximity. As illuminance drops by the square of the distance, and the virtual distance is 1/7th, then 7^2 is 49, also known as the aperture gain.

Now you can determine the amount of light per unit time upon a 7mm exit pupil by multiplying illuminance times the area:
(49 millilux or 0.049 lumen/mm^2) x (3.14(7mm/2)^2)
0.049 lumen/mm^2 x 38.465 mm^2
1.884785 lumen

Now compare the amount of light per unit time received by the eye pupil with the scope to the amount of light per unit time received by the eye pupil without the scope:
Amount of light without scope = (average illuminance from target) x (eye pupil area)
Amount of light without scope = 0.001 lumen/mm^2 x (3.14(7mm/2)^2)
Amount of light without scope = 0.001 lumen/mm^2 x 38.465 mm^2
Amount of light without scope = 0.038465 lumen

Eye Pupil Light Gain Factor = (Amount of light with scope) / (Amount of light without scope)
Eye Pupil Light Gain Factor = (1.884785 lumen) / (0.038465 lumen)
Eye Pupil Light Gain Factor = (1.884785 lumen) / (0.038465 lumen)
Eye Pupil Light Gain Factor = 49x

Because the exit pupil diameter is larger than the eye pupil diameter, the objective lens could be smaller at this magnification and still yield the same total light gain at the eye pupil. On the other hand, at 8x the full 56mm lens would be required for the same gain upon a 7mm eye pupil, as 56mm/7mm = 8. So for a person with a 7mm eye pupil diameter, this would produce both a 64x illuminance gain and a 64x total light gain.

Also as magnification increases beyond 8x with a 56mm objective lens, the exit pupil diameter becomes smaller than the eye pupil diameter, and the illuminance gain factor no longer equals the eye pupil total light gain factor. For instance at 10x, the illuminance gain factor becomes 100x and the eye pupil total light gain factor remains 64x. This is because beyond 8x magnification the 56mm lens is unable to deliver the total light gain, which is a combination of both illuminance and area. Therefore for a person with a 7mm eye pupil diameter, any magnification beyond 8x with a 56mm objective delivers no more light than 8x, just more resolution provided enough light.

As a matter of redundancy, I would prefer to just assume "per unit time" rather than keep repeating it because I don't plan on calculating light reception and perception ability over time. In how many actual applications do people really consider time when working with quantities of light anyway. It's kind of a given.

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Also, it is helpful to think of illuminance (in lux) as amps, illuminance area as volts, and total light per unit time as watts. Illuminance area and volts are like the gate, and illuminance and amps are the amounts within the gate per unit gate size.

Volts x Amps = Watts
Lux x Area = Lumen
 
Any amount of time can be applied as applicable to the calculation to complete the total amounts:
 
Volts x Amps X Hours = Watt Hours
Lux x Area x Hours = Lumen Hours
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Optics Apprentice
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Originally posted by koshkin koshkin wrote:

Wouldn't that depend on the type of the source? or are you assuming a point source?  I would probably lean toward a lambertian emitter as a more appropriate approximation.ILya

By lambertian, I assume you refer to a light source that is uniformly distributed in all directions.   While I, too, think that is appropriate, I would characterize the light we perceive as having come from a distributed reflector, not an emitter.

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