A High Resolution Telescope
The Basics : How Interference Results in Resolution Limits
Let’s start with the basics.
Diagram
1.
We have a parabolic mirror with its focal plane positioned at a distance f (the focal distance). We have two parallel rays of light originating from a common distant point source of monochromatic (single wavelength) light reflecting off of the mirror at points P1 and P2 (equidistant from the centerline). The distance from P1 to C is the same as the distance from P2 to C. Thus, the light rays from P1 and P2 arrive in phase, and result in an intense point of light at position C. So far we will only be considering those two rays of light, i.e. not the light that also hits the remaining portion of the mirror.
Diagram
2.
Light has a wave nature and, as such, not all of the light reflecting from points P1 and P2 actually arrives precisely at point C. Some of the light also arrives at every single position along the focal plane. The intensity of the light at any given point on the focal plane is determined[1] by the phase difference between the light arriving from point P1 and the light arriving from point P2. There is, therefore, some position, Z1, on the focal plane which is a distance Q1 from the centerline, at which the phase difference between the two waves of light originating from P1 and P2 is precisely one-half of the wavelength of the light. That is, the peaks (maxima) of the light waves arriving from point P1 precisely coincide with the troughs (minima) of the light waves arriving from point P2, and thus exactly cancel out one another, resulting in zero intensity of light at that position Z1. This is due to destructive interference.
Diagram
3.
If the parabolic mirror is closer to the focal plane (with a correspondingly “deeper” curvature to the mirror) then we find that the point Z1 is closer to the centerline (i.e. Q1 is smaller). As in the prior setup, Z1 is the position on the focal plane at which the distance from P1 to Z1 is exactly one half wavelength different than the distance from P2 to Z1.
Think of it this way[2]: Suppose you have two rulers, R1 and R2. You lay ruler R1 from point P1 to the centerpoint of the focal plane and you find that it’s precisely six inches. You lay ruler R2 from point P2 to the centerpoint of the focal plane, so that the two rulers intersect at that centerpoint, and you find, of course, that it, too, is precisely six inches. Now, keeping the one end of ruler R1 positioned (pivoting) at point P1 and the one end of ruler R2 positioned (pivoting) at point P2 move the intersection point of the two rulers to the left along the focal plane until the distance from point P1 to that intersection point is precisely 5 ¾ inches and the distance from point P2 to that intersection point is precisely 6 ¼ inches, that is, until the difference between the two distances is precisely ½ inch. You have just determined the distance Q1, from the centerpoint on the focal plane, at which the two distances (from P1 and from P2) are different by precisely ½ wavelength (assuming that a wavelength is one inch). If it were monochromatic, single source light that was traversing these distances, instead of rulers, from points P1 and P2, and that light had a wavelength of 1 inch, then this position Z1 on the focal plane would be the position at which destructive interference would result in a dark point.
Now, assume that points P1 and P2 were much closer to the focal plane and do the experiment again. You would find that the distance you would have to move the intersection point, Z1, away from the focal plane centerpoint in order to obtain a ½ inch (half wavelength) difference between the measurements on two rulers would be considerably less than in the prior experiment. Similarly, if points P1 and P2 were much farther away from the focal plane, you would find that point Z1 is much farther away from the centerpoint than in the first experiment.
Note that in each of these experiments we keep points P1 and P2 the same distance from the centerline of the optics. It is the distance of P1 and P2 from the focal plane that we are varying. We could also have kept P1 and P2 the same distance from the focal plane and instead varied the distance between P1 and P2, respectively, from the centerline (making certain that the distance from P1 to the centerline is always equal to the distance from P2 to the centerline).
Having performed these experiments we can draw the following general conclusions:
For a given wavelength of light and a given distance from the optical centerline to points P1 and P2:
1) The closer that points P1 and P2 are to the focal plane, the smaller the distance, Q1, between the point Z1 on the focal plane to the optical centerline.
2) The farther that points P1 and P2 are from the focal plane, the greater the distance, Q1, between the point Z1 on the focal plane to the optical centerline.
Similarly,
For a given wavelength of light and a given distance from points P1 and P2 to the focal plane:
1) The closer that points P1 and P2 are to the optical centerline, the smaller the distance, Q1, between the point Z1 on the focal plane to the optical centerline.
2) The farther that points P1 and P2 are to the optical centerline, the greater the distance, Q1, between the point Z1 on the focal plane to the optical centerline.
Diagram
4.
However, we can also think of this from a slightly different perspective. If we either (1) change the distance from our parabolic mirror to the focal plane, or (2) change the distance from the optical centerline to our points P1 and P2, we are effectively changing the angle at which our light rays from P1 and P2 approach the centerpoint of the focal plane.
So far we have always been considering the extreme marginal rays (i.e. the rays reflecting from the farthest out edge of our parabolic mirror). We can therefore say that:
The smaller the angle
that the extreme marginal rays make with the focal plane, the smaller the
distance Q1 from the optical
centerline to point Z1.
Notice that this angle is determine by the line connecting point P1 (or point P2) with the centerpoint of the focal plane. Further notice (and this is a very important distinction) that:
The points (P1 or P2) are not on
the plane defined by the distance f between the focal plane and the parabolic
mirror (identified in the above diagram as “the plane of the mirror”). Rather,
they are positioned on the surface of the mirror
itself.
We’ll discuss this distinction in more detail later on, but keep it in mind.
The Limit (well, one limit)
The minimum Q1 limit (that is, the configuration in which the distance Q1 is minimized) occurs when points P1 and P2 actually lie on the focal plane itself. In our ruler experiment, if we place P1 and P2 on the focal plane and again position or rulers to measure from those points to Z2, (the position at which the lengths differ by exactly ½ inch) we will see that Z2 resides exactly ¼ inch from the centerpoint. Extending this to the optical realm it means that the absolute minimum limit is ½ wavelength (distance from Z2 to its corresponding Z position on the opposite side of the focal plane centerpoint).
Diagram
5.
Now, recall from Diagram 1 that we had calculated the position Z1 where the difference in the lengths of lines from points P1 and P2 to Z1 was exactly ½ wavelength.
We can also note that there will also be a position Z2 on the focal plane that is a distance Q2 away from the centerpoint C, such that the distance from P1 to C is precisely one whole wavelength different than the distance from P2 to C. If we were only considering these two rays of light (from P1 and P2) then we would rightly conclude that the intensity of light at point Z2 is not zero, as it was at point Z1, but rather is a “bright” point on the focal plane. Not nearly as bright as at point C, but a bright point nonetheless.
Diagram
6.
Now, lets consider the case where we have a multitude of parallel light rays (again, all originating from a common, distant, monochromatic point source of light), reflecting off of the parabolic mirror at points +C, +B, +A, -A, -B, and -C such that point +A is the same distance from the centerline as –A, point +B is the same distance from the centerpoint as –B, and so forth for all of the points.
Diagram
6.
Below our optics diagram we now add a representation of a single wavelength of light. Remember that we are considering the situation where the distance from P1 to Z2 is precisely one whole wavelength different than the distance from P2 to Z2 (the length P2 to Z2 is one whole wavelength greater than the length P1 to Z1). Thus, the part of the wave at the left end of the wavelength representation corresponds to the part of the wave arriving at point Z2 that originates from point P1. The part of the wave at the right end of the wavelength representation corresponds to the part of the wave arriving at point Z2 that originates from point P2, which is one full wavelength behind the part of the wave arriving from point P1.
The sine wave of the wavelength represents the positive and negative energy values along the wavelength. The energy value on the wave originating from point P1 (i.e. the left-most position on the wave representation) is zero, as is its corresponding energy value on the wave originating from point P2 (the right-most position on the wave representation). Thus, the contribution of energy arriving at point Z2 and originating from points P1 and P2, is zero.
Now, consider point +C, which is 1/8th of a wavelength along the sine wave. It contributes an energy value[3] of +C to the energy arriving at Z2. But note that point –C, which is 7/8ths along the sine wave, contributes an energy value exactly the same in magnitude but opposite in sign, namely -C. Thus the energy values from points +C and -C exactly cancel one another out. Thus, together, they contribute zero energy to the point Z2. Similarly, we can see that the same principal applies to every single position between P1 and P2. Corresponding pairs exactly cancel one another out, resulting in a sum total of zero energy at point Z2.
Thus, point Z2 is dark.
If we perform the same analysis of the positions on the focal plane between the centerpoint and Z2 we will find that the centerpoint will have a very high energy level and, as we move away from the centerpoint and towards Z2 the energy level (and, hence, the intensity of the light) diminishes until it reaches zero at Z2.
Calculations for positions on the focal plane beyond Z2 will show that the energy levels increase and decrease and increase again in a simple pattern, with the distance between zero-energy points always being equal, but with the overall energy between zero-energy points rapidly diminishing. And, of course, energy levels on the other side of the focal plane centerpoint exhibit the exact same pattern.
The Airy Disc ?
The distance between the first zero-energy point to the left of the focal plane centerpoint (Z2 in our diagram) and the first zero-energy point to the right of the focal plane centerpoint is the Airy Disc size.
Well, not exactly. So far we’ve only considered a cross section of our optical device. In an actual telescope the parabolic mirror is a complete, circular mirror, and the energy level contributions from all position on the mirror surface must be taken into account when determining the distance Q2 to position Z2.
So far …we have shown that an Airy disc is formed from the reflections of light off of the entire surface of the parabolic mirror and that the size of that Airy disc is dependent upon the the angle that the extreme marginal rays, reflecting off of the extreme outer edge of the surface of the mirror, make with the centerpoint of the focal plane.
The First Equation
Hecht, and many others, correctly derive an equation[4] for calculating the radius of the first dark ring (i.e. the extent of the Airy disc), that equation being:
Equation 1
where Q is the radius of the Airy disc, a is radius of the mirror, R is the distance from the center of the mirror to point Z on the focal plane, and λ is the wavelength of light.
Diagram 7.[5]
They then go on to say:
Hecht then states that, since R is approximately equal to f (and D, the diameter of the mirror) is equal 2a, we can write:
Equation 2
But note the qualification in that statement. Hecht is saying that “since R is approximately equal to f “ we can write the equation as above. This is a very important qualification. What he is saying is that this equation, although useful in many applications, is not precisely accurate, and that if you want to be completely accurate you really should use Equation 1. Of course, in equation 1 you have the problem of actually calculating R.
A side note: From the diagrams in the book by Hecht, it is not clear whether R is measured from the centerpoint of the mirror surface or the centerpoint of the plane defined by the outer edge of the mirror. This, too, I believe is a critical distinction.
The possible
conclusion to be drawn here is that Equation 2 provides a reasonable
approximation of the Airy disc size (and hence the limit of resolution of an
optical device) for most conventional optical designs but does not
provide a truly accurate, precise equation in the general sense.
The Other
Equation
B.K.Johnson[6]
takes a different approach to deriving an equation for calculating the radius of
the Airy disc. His equation is:
Equation 3
where 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 ray
from the mirror makes with the optical centerline.
Diagram 8.
In this equation, the larger the angle U, the smaller the Airy disc size.
The values for Q (the radius of the Airy disc) that are calculated from Equation 3 are, of course, the exact same values you get from Equation 1 provided that you properly calculate the value of R in Equation 1, rather than using the “approximately correct” Equation 2.
Approaching the Limit of Resolution
Armed now with Equation 3 let’s take a look at a rather unconventional optical system. In general, for most telescopes, the radius of the primary mirror is small in comparison to the focal length, which means that the primary mirror is relatively “flat”, although it is still, of course, a parabolic curve. But let’s look at a telescope design in which we use “more” of the parabolic curvature; in which the focal length is nearly equal to the mirror diameter:
Diagram 9.
In this
configuration we get a very small Airy disc which, of course, means exceptional
resolution. In fact, it attains a resolution better than any conventional
telescope. And, since we’ve shown that resolution is determined by the angle U,
and not by the ratio of diameter to focal length, as is implied by the
“approximation” equation 2, this configuration can be used for mirrors of any
size, even very small sizes, while still remaining exceptional
resolution.
Conclusions
Although the relationship between mirror size and resolution, as expressed in Equation 2,
is appropriate for most common telescope configurations, in which the mirror diameter is small relative to the focal length, it is not a precise equation. As noted by Hecht, it involves an approximation; namely that R, the distance from the center of the mirror to point Z on the focal plan (per Diagram 7 and Equation 1) is approximately equal to f (the focal length). Thus, the equation itself is only an approximation, and for telescope configurations in which the mirror diameter is medium or large sized in relationship to the focal length (i.e. in which the mirror is a “deep parabola”)[7] the equation is a very poor approximation.
On the other hand, the equation presented by B.K. Johnson,
is a very precise description of telescope resolution, and applies equally well to all telescope designs.
This equation shows that a telescope design such as that shown in Diagram 9 will provide exceptionally high resolution, regardless of the actual diameter of the primary mirror. Indeed, a six inch mirror with an appropriate “deep parabola” and relatively short focal length should exhibit exceptional resolution.
RoadRunner ?
A question: Can a file be set up for RoadRunner to incorporate a “deep parabola” to determine the validity of these conclusions? That is, does RoadRunner actually simulate the interference of the light waves coming from the mirror surface or does it merely use equation 2 to calculate resolution?
On To Angular Resolution
So far what we’ve shown is that even a small diameter mirror can attain high resolution. However, the resolution that has been discussed thus far relates only to the resolution of the image at the image plane. That is, we’ve defined the size of the Airy disc as it appears at the image plane. This is comparable to defining the size of the ink “dot” that appears on a printed page, but it does not tell the whole story.
Diagram 10.
In Diagram 10 we have a simple telescope with a particular focal length (f), and we’ve shown two Airy discs, each one corresponding to a single, distant, point source of light, such as a star, that are just overlapping at the center of the focal plane. (The Airy discs are shown “facing us” so that we can see their size and position, but obviously in reality they would be flat against the focal plane.)
Now, let’s consider the Airy disc A in Diagram 10 and ask the question,
‘Where in the sky, relative to the telescope, must the point source of light be in order for it to result in Airy disc A being positioned exactly where it is on the telescope’s focal plane?’
Diagram 11.
We can answer this question by tracing te light path back from Airy disc A, to the centerpoint of the mirror surface, and then back out into space, to Star A.
Diagram 12.
Similarly, we can trace Airy disc B back to its source, Star B. And having done so, we can see that the two just-distinguishable Airy discs on the focal plane effectively define an angle that we’ll call the Angle of Resolution. Or, we can say that the Airy disc size defines the Angular Resolution of the telescope.
Diagram 13.
If another point source of light, Star C, happens to reside between Stars A and B, we can see that it’s image (i.e. its Airy disc) on the focal plane will be significantly overlapped, and thefeore indistinguishable, from the Airy discs of Stars A and B. Thus, it will not be possible to resolve the image of Star C. Only point sources of light that are separated by at least the Angle of Resolution will be distinguishable from one another.
So, we can now finally see why the limits of resolution of a telescope are determined by two factors: (1) the Airy disc size (which is determined by the angle at which the extreme marginal rays from the mirror surface make with the focal plane, which in turn is determined by the focal length, the mirror size, and the mirror curvature), and (2) Angular Resolution (which is determined by the focal length and the Airy disc size). It should be obvious that, for a given Airy disc size at the focal plane, the longer the focal plane the smaller the Angular Resolution. Thus, a long focal plane is desired in order to attain the best images with the most detail.
Diagram 14.
Now, let’s return to our “Deep Curvature Mirror” which, as we saw previously, has an incredibly small Airy disc size, i.e. a very high resolution at the focal plane. And let’s suppose that this is a fairly small mirror, say, 6 inches in diameter. In this case, even though we have a very high focal plane resolution, the focal length is relatively short, in fact extremely short. This would mean that, even though we have high resolution at the focal plane, we have low Angular Resolution, and thus not very good image detail.
Is there anything that can be done about that?
Diagram 15.
Suppose that we place a magnifying lens or set of optics in front of the telescope, effectively making the telescope “look” at a narrower field of view. The “effective” Angle of Resolution becomes much narrower, meaning finer detail at the focal plane.
Or, …
Diagram 16.
Perhaps we can take an even more drastic approach, turn the primary mirror upside down, and add a magnifying off-axis mirror. Again, we attain a much narrower effective Angle of Resolution and, hence, finer detail at the focal plane.
It would seem, then, that, armed with a deep dish primary mirror with exceptionally high resolution at the focal plane, we can attain exceptionally high angular resolution by merely adding magnifying optics in front of the primary mirror. Thus, even a small telescope of this design should be able to attain image quality and angular resolution comparable to extremely larger conventionally designed telescopes.
Yes?