Interesting theory but this thread from Cloudy Nights seems to disagree that objective size can increase contrast ratio. I am not totally convinced because I feel contrast is better in bigger binoculars and scopes. Here is the discussion:
An excellent exploration of the eye pupil vs. exit pupil issue. That said, I would like to comment on the following paragraph from your article:
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What consideration should be given to exit pupil for use in brightly lit light polluted skies?
In this case you might consider even smaller exit pupil than your maximum dilated eye pupil. You can vary the values in the relationship of aperture/magnification (which gives exit pupil) by either decreasing aperture or increasing magnification. Either choice will decrease exit pupil and therefore increase the apparent contrast in the image by decreasing the brightness of the background sky. This should help bring out the images of target objects.
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Unfortunately, contrast ratio (see below note) is always at its maximum to the naked eye. You can't improve the contrast ratio using any optical aid at any magnification.
Put simply, Contrast ratio is the ratio of object surface brightness to sky surface brightness. This ratio of surface brightnesses determines the contrast between the object and background sky. Surface brightness is defined as a brightness per unit area, and is typically expressed by astronomers in units of magnitude per square arcminute or magnitude per square arcsecond.
Surface brightness is determined by dividing the total brightness of the object into its area. For the sake of discussion, let's say an observer is using his 60-mm refractor to look at some distant galaxy. To the naked eye, the galaxy has a surface brightness, SBgal, and the sky has a surface brightness, SBsky. If the observer has 6-mm eye pupils, let's find out how the telescope affects the contrast ratio of the galaxy vs. background sky.
The telescope has an aperture (60-mm) 10x that of the observer. As a result, the telescope theoretically collects 100x as much light. But to take full advantage of that aperture, the observer needs to use at least 10x magnification in the telescope. 10x produces a (60/10=6) 6-mm exit pupil in that aperture, which just happens to match the observer's eye pupil size.
So, the telescope collects 100x as much light as the observer. But the telescope also must spread that light over (10x^2) 100x the surface area in producing the largest exit pupil that preserves the full aperture of the instrument. In short, the surface brightness of the galaxy (SBgal) remains unchanged. And since the background sky behaves just as any extended DSO, its surface brightness (SBsky) also remains unchanged. The net result is that contrast ratio also remains unchanged by the application of larger aperture.
What happens if we use higher magnifications to lower the surface brightness of the sky? The extended galaxy's surface brightness is reduced by an equal amount. And again, we find that the ratio of their respective surface brightnesses (SBgal/SBsky) remains unchanged.
What happens if we use even lower magnifictations that produce exit pupils larger than the eye pupil? Well, the apparent surface area of the object is decreased. On the surface, this may seem a good way to increase object surface brightness. However, as you clearly demonstrate in your article, the larger exit pupil effectively reduces the aperture of the instrument. The net result is that object surface brightness is lowered due to the reduced effective aperture. And contrast ratio is, again, unchanged.
But by reducing the effective aperture, we've also raised the contrast threshold at which objects become visible to the eye. (For a discussion of aperture, sky brightness and contrast threshold, please see the following: ODM Matrixes) So, even though contrast ratio is unchanged, we've actually made it harder to detect the object. Some objects that were visible at the threshold are now invisible in that reduced aperture.
The inescapable conclusion is that you cannot improve contrast ratio by the application of increased aperture, or by changing magnifications and, by extension, exit pupil. To illustrate, let's look at the scenario you discussed in the above quoted paragraph; the observer in a light-polluted environment choosing smaller binoculars.
I own a pair of 10x50s. If I brought those and a friend's 10x25s into the field, what could I expect in terms of performance? The 10x50s produce a 5-mm exit pupil. The 10x25s produce a 2.5-mm exit pupil. Both magnify the target by 10x, so the target has the same apparent surface area in both binocs. For the sake of discussion, I'll assume my eye pupil dilates to 5-mm under these conditions. Since the 50-mm binoculars have twice the aperture, they collect 4x as much light. Therefore the surface brightness of the target object will be 4x greater in the 10x50s than in the 10x25s.
Clearly, there is no advantage to using the 10x25s in this situation. It's been shown that the contrast ratio is the same in both instruments. Object apparent size is the same. The only binocular with an advantage is the 10x50, which presents objects as having 4x the surface brightness vs. their appearance in the 25-mm aperture binocs.
Suppose that, instead of bringing my friend's 10x25s, I bring his 20x50s into the field. How would these stack up against the 10x50s? Well, the 20x50s collect the same amount of light but produce a 2.5-mm exit pupil. Object surface brightnesses would be reduced but that wouldn't impact the contrast ratio. I'd need to use the higher mag binocs on a tripod or some sort of stabilizing device to address the shakes. And if we're talking about a very bright observing location, my eye pupils may not open wider than about 2.5-mm. The 20x magnification would, in that scenario, allow me to make full use of the aperture. The 10x50s would not.
If my eye pupils are just 2.5-mm in size, then the 10x25 binocs should--all other factors being equal--perform just as well as the 10x50s. That's because, as you've shown, my eye pupil shuts down the 10x50s to a 25-mm effective aperture. Theoretically, both instruments perform as 10x25s.
But the smaller binocs wouldn't have any optical advantage.
In conclusion, I'd recommend you consider editing your article to reflect the fact that contrast ratio (object surface brightness vs. sky surface brightness) is not affected by changes in aperture or magnification. Two ways to improve contrast ratio, are to observe under a darker sky or to use line filters that reduce sky brightness more than object brightness. Increasing aperture has the effect of lowering contrast threshold, which produces the perception of improved contrast ratio.
Although I agree with Henry that we should not jump to conclusions about aperture, I'm still tempted to suggest something on the basis of what Ron's curves as well as calculated MTF curves for diff.-lim. optics show. Along the lines of what Ron says in the following quote from post #78:
"One thing I’ve noticed right off, if the stab is close to correct, is that it appears larger objectives are going to have a contrast advantage over the smaller objectives even at the lesser resolutions."
The point I wish to make is that much of what we view in daylight involves trying to make out detail that is not of particularly high contrast to begin with - much different from a black-and-white bar-target or a double-star against dark sky background. So, supposing my eye is dilated to anything bigger than the 2.5mm an 8x20 binocular offers, there perhaps is a meaningful difference brought by increased aperture in my ability to extract detail from all sorts of lower-contrast objects where the spatial frequency/contrast ratio of the target was around the detection threshold of my eyes.
The personal reason why I keep coming back to this is that in addition to the limited testing I have done, I have now for some three years owned and used a very good sample (very low aberrations, excellent measured resolution for the aperture) of the Leica Ultravid 8x20, and am consistently surprised by how seldom I find the view fully satisfying despite knowing full-well there is very little room for improvement in it within its size and design parameters.
Kimmo
Kimmo