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Telescope Resolution
How a telescope can be built which exceeds the
“Limit of Resolution”

It is well known among telescope designers, both amateur and professional, that “aperture defines resolution”. This means that the resolution capability of the telescope is determined by its aperture, i.e. the front end opening of the telescope. Unfortunately, this is wrong.

The first thing to correct about this misconception is the clarifcation of what constitutes the aperture of the telecope. To be precise, the aperture is the area of the primary mirror that reflects light to the focal plane. In general it is the entire surface area of the primary mirror (minus a central portion of the mirror that is blocked by the receptor at the focal plane or a “diagonal” that ...) that constitutes the aperture and it is measured as the radius (or diameter) of the primary mirror.

It is fairly easy to see why the light-gathering capability of a telescope is determined by the size (aperature) of the primary mirror; the larger the mirror the more light it reflects towards the focal plane. This means that the image formed at the focal plane is brighter for larger telescopes. It also means that fainter and more distant objects (which are implicitly fainter due to their distance) can be imaged more easily (with shorter exposure times). But it is not so clear as to why the resolution of the telescope is also determined by its size.

Some Basics : The Airy Disc

When a far distant point source of light is focused by a telescope onto its focal plane one might expect for it to result in a precise point of light at that focal plane, but such is not the case. It would be the case if light were to act solely as particles, or photons, of light, in which case we might expect the point at our focal plane to be, at a minimum, the size of a single photon. But light also has a wave nature, i.e. it behaves as if it arrives at the focal plane as a series of waves. And, as anyone who has ever watched ripples on the surface of a pond knows, waves interact with one another. If two light waves arrive at a given point on the focal plane at exactly the same time so that their maximums (peaks) and minumums (troughs) arrive at the same time (i.e. completely synchronized) then the intensity of  the point of light at that particular position experiences what is called “constructive interference” and we get a very bright point of light. On the other hand, if these two waves arrive “out of synch” such that their respective peaks and troughs do not precisely line up, then it experiences “destructive interference” and the intensity of light at that point is somwhat less then maximum. In the extreme case where the waves are precisely one-half wavelength out of synch, it experiences complete destructive interference and the intensity is zero (black).

This results in not just a single larger-than-expected spot of light at the focal plane (for that single, far-distant, point-source of light that we were considering) but rather results in a central spot surrounded by a (theoretically infinite) set of successively dimmer rings  of light around that central point. On a well-designed telescope the overwhelming majority of light resides within the central point, but some of the light always resides in those additional rings of light.

Since the rings of light surrounding the central point normally (for a well-desighned telescope) are considerably dimmer than the central point of light they are generally ignored when considering the “size” of the point of light.

The radius of that central point of light can be determined by the following equation:

    R = (1.22 * w) * (f   / D)

where f is the focal length of the primary mirror, w is the wavelength of the light, and D is the diameter of the primary mirror,

Actually, a more precise equation is given by:

   R = (1.22 * w) /  (n SIN(u))

where w is the wavelength of the light, n is the refractive index of the medium in the image space (which is normally air and therefore equal to 1), and u is the angle that the extreme marginal rays from the mirror makes with the optical centerline.

These two equations should make it clear that the Airy disk size is determined by the angle at which the rays of light strike the focal plane. Said another way, it is the ratio between the distance from the pirmary nirror to the focal plane and the diameter of that primary mirror. The important point here is that the Airy disc size is not determined by the actual size of the mirror or its focal length but rather by the ratio between those two measurements. This means that a small telescope with a 5 inch diameter mirror and a 20 inch focal length will have the exact same Airy disc size as a large telescope with a 5 foot diameter mirror and a 20 foot focal length, because the ratios between focal length and diameter are the same.

The bottom line here is that the closer the ratio of focal length to diameter is to 1 (or, equally, the closer the SIN of u is to 1), the smaller the Airy disc size, and the smaller the Airy disc size the better the resolution of the telescope.

A more detailed explanation of the Airy disc can be found here.

So, why does “Bigger mean better resolution” ?

More Basics : Angular Resolution

The Airy disc size, however, is not the full determination of the resolution of a telescope. It is crtical to resolution, to be sure, and the smaller the Airy disc size the better the resolution, but there is one more item to consider, and that is Angular Resolution.

Suppose that we’ve got a telescope that produces a nice tiny Airy disc for each distant, point source of light (star). And suppose that we point the telescope up into the night sky and find that we get exactly two Airy discs (A and B) at our focal plane and that these two Airy discs are just far apart enough to distinguish one from another. That is, they overlap a little bit but we can clearly see that there are two distinct Airy discs, and therefore that they correspond to two separate stars in the sky. If they were any closer together they would overlap too much and we wouldn’t be able to correctly determine that there were indeed two separate point sources of light.

We could then trace two lines, one for each Airy disc (A and B) at the focal plane, back through the telescope’s mirror, and out into space to its respective source star.Thus, we would find the two stars that result in the two just-distnguishable Airy discs at our telescope’s focal plane. If there were some additional star, C, between the two stars A and B then its (i.e. C’s) Airy disc would appear at the focal plane between the two Airy discs from the stars A and B. But since Airy discs A and B are so close that they are just-distinguishable form one another, the Airy disc of star C would be “lost” (would be indistinguishable) in the overlap of the Airy discs.

What this means is that our telecope’s resolution is such that it can just barely resolve between stars A and B, because they are far enough apart (out in space) to result in just-distinguishable Airy discs at the focal plane of the telescope.

If we look at those two lines that we traced from the Airy discs back to their respective stars we’ll see that there is an angle between those two lines. This angle defines the Angular Resolution of our telescope, and it is the true measure of the resolution capability of our telescope. The smaller the angle, the better the resolution.

This angle is determined by two things, the Airy disk size and the focal length of the telescope. Since we have previously shown that the Airy disc size is not determined by the actual size of the telescope (but rather by the ratio of focal length to mirror diameter), this means that, for a given Airy disc size (i.e. a given ratio of focal length to mirror diameter), it is the actual size of the focal length that determines the Angular Resolution. The longer the focal length, the smaller the Angular Resolution.

Thus, when we say “Bigger is better” it doesn’t really mean a bigger mirror but rather a bigger focal length. Since most “big” telescopes are big both in the mirror diameter and the focal length it’s sort of, kind of, OK to say that bigger mirrors means better resolution, but it’s not technically correct.

In fact, if it were possible to construct a small diameter mirror telescope with a very long focal length, you could have exceptional resolution in spite of the fact that the mirror diameter is small. But can that be done?

Small Diameter, Long Focal Length = Exceptional Resolution

Let’s summarize what we know so far:

1. The Airy disc size is determined by the angle at which the extreme marginal rays (from the primary mirror) approach the focal plane, relative to the optical centerline. This means that the Airy disc size is determined by the ratio between the focal length of the primary mirror and the diameter of the primary mirror (assuming that the entire mirror surface contributes to the image formed at the focal plane, except for that small central portion obscured by the imaging device at the focal plane).

 

 

 

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