glenn, jon, i appreciate the clarifications and generous spirit.

i tried glenn's pinhole experiment (as jon said, it's difficult to do, i punctured a foil sheet many many times and bobbed my head a little to get star image, exit pupil, foil hole and eye all collimated), and found paradoxical results sighting on vega with a 180mm Æ’/15 mak cass: stopping down a low power eyepiece (EP of around 1.6) the image became quite dim and a small airy disk became visible, which may have been too faint to see completely; but stopping down a high power eyepiece (EP of 0.33) produced very little dimming and no perceptible (clearly different) change in the size of the airy disk. i assume this is because the eyepiece EP and the foil hole were both around 0.33.

if the exit pupil, the most variable part of the visual system, is equivalently a criterion for system or image resolution, then why not use it as such? you'd calculate exit pupil directly from Æ’(eyepiece)/N, then calculate resolution as 206265*(lambda/EP) and magnification as D/EP. we would not use aperture (entrance pupil), D, because it is automatically "stopped down" in all situations and therefore the tail end of the system. the resolution on the image plane (aperture diameter) would be superfluous, if we are estimating the *system* resolution. that is probably the crux of my confusion.

if that procedure is incorrect, then we're back to my original statement that stopping down the aperture (at the front) is not the same as stopping down the exit pupil (in the back), because as glenn pointed out in the normal range of the human iris, it has no discernable visual effect, other than reducing the image illuminance, which is equally well and factually explained by the simple statement that "you've stopped down the exit pupil". calling in the aperture is to my ear an empty rhetorical gesture. if it doesn't matter where the stopping down occurs, why say it occurs other than where it actually does?

Bruce:

Here's some things to consider:

The exit pupil itself is not a measure of the system's angular resolution, the exit pupil multiplied by the magnification is a measure of the system resolution, the exit pupil multiplied by the magnification is equal to the aperture. One can work backwards from the Rayleigh criteria and show that angular resolution ® = 5.45 inch/(Mag x Exit pupil)

The issue here though is somewhat more fundamental. When the exit pupil is larger than the entrance pupil, it means that the entire objective is not seen and that light from the outer portion of the objective does not enter the eye, the aperture is effectively stopped down. So, the fundamental question here is whether light that is at the focal plane but does not enter the eye can increase the resolution, that it somehow decreases the size of the Airy disk and increases the resolution even though it does not enter the eye. I think the key here is to realize that light passing through an aperture causes diffraction effects that increases the size of the Airy disk. If the light passes through multiple aperture stops, then the overall size of the disk depends on the both apertures and not just the first one. Thus when the entrance pupil is smaller than the exit pupil, not only does one have to consider the aperture imposed by the objective but also the aperture imposed by the undersized entrance pupil.

One way to see this effect is to calculate Rayleigh criteria (R= (138/D) mm-arcseconds) for the angular resolution of the telescope and the angular resolution of the eye. Consider the 100mm telescope at 100x, First I will calculate the angular resolution of the telescope and then the angular size of the Airy disk:

Rscope = 138mm/100mm = 1.38 arc-seconds. Magnify that 100x to produce a 1mm exit pupil and it will be 138 arc-seconds. That is the angular size of the Airy disk in a 100mm telescope at 100x.

Now consider the resolution of a 1 mm lens, the eye at a 1 mm exit pupil:

Reye1mm = 138mm/1mm = 138 arc-seconds, This is the same as the telescope, the light passes unimpeded because the entrance pupil is at least as large as the exit pupil and the entrance pupil imposes no further diffractive effects.

Now consider what happens when the entrance pupil is only 0.5mm even though the exit pupil is 1mm.

Reye.5mm = 138mm/.5mm = 276 arc-seconds, the aperture of the eye produces an Airy disk that is twice the diameter of the angular resolution of the telescope, the resolution of the system is now reduced to the resolution of a 50mm telescope rather than a 100mm.

I could generalize these calculations and use algebra to show that the size of the Airy disk is determined by the effective aperture of the telescope just as the light gathering is but I think this example should suffice to show that in terms of resolution, stopping down the aperture at the entrance pupil is equivalent to stopping down the aperture . One only needs to know the effective aperture of the telescope to determine the resolution.

Something interesting to understand: The magnified angular size of the Airy disk in a 1mm exit pupil is independent of aperture, larger apertures require proportionally larger magnifications to produce the exit pupil so while the actual resolution and angular size of the airy disk at the focal plane are smaller in the larger scope, they are independent of aperture at a given exit pupil. A 200mm scope at 200x produces a magnified airy disk that is the same diameter as a 100mm scope at 100x and it's the same angular size as the 1mm exit/entrance pupil at 1x and half the size of a 0.5mm exit/entrance pupil at 1x.

All this is not so easy to understand, for those reading it, please take the time to read it and reflect before responding. It took me quite a while to put this into a concise form, there maybe grammatical and mathematical errors but the concepts are solid and the conclusion that the resolution of depends on the effective aperture and it does not matter whether the aperture stop at the objective, the entrance pupil or somewhere else is solid.

I will say I clarified some issues in my mind working this through, I hope others have gained understanding as well.

One final thought: These considerations, while fundamental to understanding what is going optically, have very little importance from a practical point of view. This is because the entrance pupil of the eye is never small enough to mask exit pupils that are small enough for the Airy disk to be seen. One's eye may only be open to 4mm but no one sees Airy disk structures in a 4mm exit pupil, much higher magnifications are needed, something close to 1mm and the entrance pupil of the eye is always larger than 1mm. At the magnifications where resolution of the telescope is important, the entrance pupil will always be larger than the exit pupil. At large exit pupils, the resolution/response of the eye is such that masking the aperture so the effective aperture of system is reduced can actually increase the eye's ability to resolve double stars. I once did a test on gamma Arietis, a 8 arc-second double with an 80mm refractor at a constant 17x magnification, a 4.7mm exit pupil. Keeping the magnification constant and masking the aperture increased the apparent resolution.. There are many factors to consider but a simple focal ratio analysis is useful. the eye has a focal length of about 17mm so a 4.7mm exit pupil means the eye is operating at F/3.6.. ,not likely to be too sharp...

'nuff said.

Jon Isaacs