Angular-resolution concept

What exactly is angular resolution? It’s the minimum angle over which two points may be seen as separate rather than blurred together. Those two points may be two separate light sources, such as two stars, or they may be details on a single object, such as the edge of the Red Spot on Jupiter. So with bet-ter (smaller) angular resolution, you’re able to see objects more clearly. If you wear eyeglasses or contact lenses, you can easily demonstrate the benefits of better angular resolution by comparing your view of the world with and with-out your lenses. Withwith-out optical aid, your eyes present you with a view of the world that is fuzzy and in which details are unresolved.

Resolution is related to the wave nature of light, and to understand resolu-tion, you need to understand how waves interact. When two or more waves are present at the same location, the interaction between those waves is called

“interference.” And although interference has a negative connotation in every-day use, in science interference may be constructive, destructive, or something in-between. A few examples of interference between two waves are illustrated in Figure 4.7.

Notice that when the two waves are in step (also called “in phase”), they add constructively to produce a larger wave. But if those same two waves are out of step, they add destructively to produce a smaller wave (which may be no wave at all, if the two waves are perfectly out of phase and equal in size). And, if the waves are just slightly out of step, the resultant wave is not as big as the perfectly in-step case, but it is still bigger than either of the constituent waves.

To see why wave interference is relevant to angular resolution, you have to consider the waves gathered and brought together by a lens or mirror. In Figure 4.8, you can see a slice through a lens and the effect of the lens on incoming light waves.

112 Parallax, angular size, and angular resolution on lens axis - all waves in step at this point Figure 4.8 Waves passing through a lens.

Notice that in this figure, the incoming waves on the left are all parallel to one another; this is due to the very great distance to source. For a closer source, the waves would be diverging (getting farther apart as they travel), but even the closest astronomical objects are so far away that their waves are essentially parallel by the time they get to Earth.

As you can see in this figure, after passing through the lens the light waves converge toward a point (the angles are exaggerated for clarity). Since the light source in this figure is on the axis of the lens (which is the line passing through the center of the lens and perpendicular to the lens), the “focal point” (the point to which the waves converge) is also on the lens axis. Waves coming from other directions will focus to different points, and the “focal plane” is the locus of all the points to which the waves converge.

It’s important to understand that at the focal point shown in Figure 4.8, waves from all points on the lens (center, edges, and in-between) are all in step. That means that these waves will add constructively at this location, pro-ducing a bright spot on the image. But even if the source of light is a point

4.3 Angular resolution 113 along focal plane

Brightness

Figure 4.9 Brightness near the focal point.

(near-zero angular size), the waves add constructively not just at a single point on the focal plane, but over a small region. You can get an idea of this by look-ing at the little sideways graph on the right side of Figure 4.9, which is a plot of the brightness on a slice through the focal plane.

That strange-looking graph with a large central peak surrounded by lots of little bumps is called the “point-spread function” (PSF) because it shows how the light from a single source point is spread out on the image. Notice that the central peak is not infinitely narrow; it has finite width even if the light waves come from a single point. You should also notice that there are a series of “nulls” (points of zero brightness) between the minor peaks on both sides of the central peak.

The width of the main peak and the location of the nulls and minor peaks depend on two things: the size of the lens and the wavelength of the light. For a given wavelength, bigger lenses produce narrower peaks and smaller lenses produce wider peaks. And for a given lens, longer wavelengths produce wider peaks and shorter wavelengths produce narrower peaks. To understand why that’s true, look at Figure 4.10.

As shown in this figure, at the point of maximum brightness, waves from all points on the lens add in-phase. Moving a small distance away from the focal point causes the waves from the edge of the lens and the waves from the center of the lens to be slightly out-of-step, so they add to a smaller value of bright-ness. Moving farther from the focal point causes the wave to get increasingly out-of-step, so the value of the brightness gets smaller. Eventually, if you move far enough from the focal point, the edge and center waves are completely out of step (out of phase by 180^◦), so they cancel. The canceling waves produce zero brightness, so the PSF has a null at this location. As you continue moving away from the focal point, some of the waves get back in phase, but others are out of phase, producing a series of minor peaks. You can see a detailed

114 Parallax, angular size, and angular resolution

All waves in step, add to maximum brightness Brightness

Position on focal plane

Edge and center waves slightly out of step, add to smaller brightness

Edge and center waves completely out of step, add to zero brightness Waves from some portions of lens in step, add to small but non-zero brightness

Figure 4.10 Example of a PSF.

Rotating the PSF

produces this three-dimensional function

which makes an image like this on the focal plane These nulls correspond

to these dark circles

Figure 4.11 Relationship between PSF and image.

analysis of lens operation in an Optics book, but the important concept for you to understand is that even point sources don’t focus to a true point – the light is spread over a small region. The bigger that region, the “fuzzier” the image looks.

To understand how to relate the graphs of the PSF to the image you see when you look through a telescope, take a look at Figure 4.11. Since the PSF graphs shown in Figures 4.9 and 4.10 represent a single slice through the focal plane produced by one slice of the lens, the entire image produced by a circular lens can be viewed as the combination of many such slices, each taken at a different

4.3 Angular resolution 115

Lens axis Edge w

ave Edge wave Telescope

lens

Waves from very distant point source

above lens axis All waves in step

at this point below lens axis Focal plane

Graph of brightness along focal plane

Figure 4.12 Waves from a point source above the lens axis.

angle. The rotated PSFs combine to produce the brightness function shown on the right side of Figure 4.11, which projects onto the focal plane as a bright spot surrounded by concentric rings. The spot in the middle corresponds to the bright central peak, the light rings correspond to the minor peaks of the PSF, and the dark rings correspond to the nulls of the PSF. This ring pattern is called the “Airy pattern” produced by the lens from a point source, and the bright central spot is called the “Airy disk.”

So what do PSFs and Airy patterns have to do with angular resolution? To understand that, consider the waves coming from a source slightly above the lens axis, as shown in Figure 4.12.

Notice that in this case the focal point at which the waves add in phase is below the lens axis, and the peak of the PSF is shifted downward relative to the on-axis case of Figure 4.9. Likewise, for sources below the lens axis, the peak of the PSF appears above the lens axis on the focal plane.

Now consider what happens when the waves from two sources strike the lens at the same time, as shown in Figure 4.13.

In this figure, the wave directions are indicated by straight lines (called

“rays”) to make it easier to show waves from two directions striking the lens.

As you can see in the figure, the two sources produce two PSFs, which may overlap (depending on the angular separation between the sources). Figure 4.14 shows two PSFs and the corresponding Airy disks on the focal plane. In this case, the peaks of the two PSFs are sufficiently separated to show that two separate sources exist – these two sources are said to be “resolved.”

But consider a situation in which the angular separation between two sources is small enough so that their PSFs overlap significantly – not just in the minor peaks and nulls, but in the main peaks as well. If the peaks of the PSFs overlap

116 Parallax, angular size, and angular resolution

Telescope lens Rays from source

above axis Focal

plane

Graph of brightness along focal plane

Overlapping Rays from source PSFs

on axis

Figure 4.13 Two sources producing overlapping PSFs.

PSF of second source PSF of

first source

Figure 4.14 The PSFs and images of two sources.

so much that it’s impossible to tell if there are two separate sources or one extended source, the sources are not resolved.

Exactly how much overlap between the peaks of the PSFs can be tolerated before the two sources become indistinguishable from a single, larger source?

Several criteria exist for determining whether two sources are resolved, but most common is the Rayleigh criterion. To meet this criterion, the separation between the peaks of the two PSFs must be at least as great as the separation between the peak and the first null of a single PSF, as shown in Figure 4.15.

Meeting the Rayleigh criterion ensures that there is a small dip in brightness between the peaks, and an observer can recognize that there are two separate sources rather than a single extended one. You can see an example of an image of two sources just resolved by the Rayleigh criterion in Figure 4.16.

4.3 Angular resolution 117

Distance between peaks

Distance from peak to first null

PSF of first source

PSF of second source

Figure 4.15 The Rayleigh criterion for resolving two sources.

Two sources with angular separation

greater than Rayleigh criterion

Two sources with angular separation

equal to Rayleigh criterion Figure 4.16 Images of two resolved sources.

The final concept you need to consider before you can understand the equa-tion for angular resoluequa-tion based on the Rayleigh criterion was menequa-tioned above: the width of the PSF depends on the wavelength of the light and the size of the lens or mirror. For any given wavelength, the larger the lens, the narrower the PSF. So a telescope with a large aperture produces a narrower PSF than a telescope with a smaller aperture, as shown in Figure 4.17.

The reason for this is that larger lenses have greater distance from the cen-ter to the edge of the lens, and the greacen-ter that distance, the less angle it takes to cause the edge waves to get out of step with the center waves. And since bigger lenses produce narrower PSFs, the angular resolution of a big lens is better (that is, smaller) than the angular resolution of a small lens. This

118 Parallax, angular size, and angular resolution

Figure 4.17 Point-spread functions of large and small lenses.

means that the angular resolution of a telescope is inversely proportional to the aperture:

angular resolution∝ 1

aperture, (4.9)

where “aperture” is the edge-to-edge size of the lens or mirror.

The width of the PSF also depends on the wavelength of light used, because the shorter the wavelength, the smaller the angular shift it takes to make the edge waves get out of step with the center waves. This means that the angular resolution is directly proportional to wavelength (λ):

angular resolution∝ λ. (4.10)