Solution

We use a diameter D = 5 mm for the pupil. We use equation (4.1a) to find the angular resolution in radians. We convert from radians to arc seconds to

H Diffraction. (a) A light ray enters from the bottom, and passes through a slit of length D. Diffraction spreads the beam out and it falls on a screen.The intensity as a function of position on the screen is shown at the top. Most of the energy is in the main peak, whose angular width is approximately A/D (in radians). Smaller peaks occur at larger angles. The effect in a real image. (b) [ESO]

H Diffraction. (a) A light ray enters from the bottom, and passes through a slit of length D. Diffraction spreads the beam out and it falls on a screen.The intensity as a function of position on the screen is shown at the top. Most of the energy is in the main peak, whose angular width is approximately A/D (in radians). Smaller peaks occur at larger angles. The effect in a real image. (b) [ESO]

convert the result to a convenient unit (1 rad 2.06 X 105 arc sec; see Example 2.3).

The eye's resolution is not quite this good, since the full diameter of the pupil is not generally used.

From equation (4.1a) we can see that we can improve the resolution if we use a larger aperture. A larger telescope will give us better resolution. A 10 cm diameter telescope (20 times the diameter of the pupil of the eye) will give an angular resolution of 1 arc sec. However, diffraction is not the only phenomenon that limits resolution. The Earth's atmosphere also distorts images.

When light passes through the atmosphere from above, it is passing through increasingly dense air. As the density of air increases, its index of refraction increases. Therefore, the light encounters an increasing index of refraction as it passes through the atmosphere. We can think of the atmosphere as having a large number of thin layers (as shown in Fig. 4.2) each with a slightly different index of refraction. As the light passes from one layer to the next it is bent slightly towards the vertical. The star appears to be higher above the horizon than it actually is.

This would not be a problem if the atmosphere were stable. However, variations on time scales shorter than a second cause changes in the index of refraction in some places. The image moves around. If we take a picture, we just see a blurred image. This effect is called seeing and usually limits resolution to a few arc seconds. We refer to the numerical value of the blurring as 'the seeing'. At a good observatory site, on a good night, the seeing might be as good as 1/3 arc sec or better. This corresponds to the diffraction limit of a 30 cm diameter telescope. Building a larger telescope does not help us past the seeing limitation on resolution, but it improves the light-gathering power. Hence our earlier statement that light gathering is the main purpose of large ground-based optical telescopes. We will also see later in this chapter that there are techniques for overcoming the effects of seeing to produce dif-

Ground

Image Moves Around

Image Moves Around

Seeing. (a) Bending of a light ray as it passes through the atmosphere.We can think of the atmosphere as being made up many thin layers, each with a slightly larger index of refraction as you get closer to the ground. The amount of bending is actually much less than in this picture. (b) Effect of changes in the amount of bending on the image of a star.

fraction limited images for telescopes with diameters from 1 to 2 m.

4.1.3 Image formation in a camera

To illustrate some basic points about the formation of images in optical systems, we look at the operation of a simple camera (Fig. 4.3). For astronomical situations, we are dealing with objects that are 'at infinity', so the light rays from a point on the sky are traveling parallel to each other. In the figure, we show bundles of rays coming from two different stars. The rays within each bundle arrive at an angle with each other equal to the angular separation of the stars on the sky.

For a camera with a lens of focal length f, the rays in each bundle are brought together at a

distance f behind the lens. (The image is one focal length behind the lens when the object is at infinity. That is the definition of the focal length.) The images of all the stars in a field lie in a plane, called the focal plane. The images of two stars are at different points in the focal plane. We can locate the image of each star by following the chief ray of each bundle (the ray that passes through the center of the lens, undeflected) until it intersects the focal plane.

If stars have an angular separation 0 on the sky, then, as viewed from the lens, the two images have an angular separation 0 on the focal plane. This is simply the angle between the two chief rays. The camera provides no angular magnification. As viewed from the lens, the angular separation between the stars is the same as the angular separation of the images.

We can also find the linear separation x between the two images. From the right triangle in the figure, we see that tan (0/2) = x/2f

If 0 is small, then tan (0/2) is approximately 0/2, in radians. This gives us

Solving for x gives x = f0

This tells us that the linear size of the image is proportional to the focal length. To obtain a larger image, we use a longer focal length lens. (This is what we are doing when we put a tele-photo lens in a camera.)

Apart from image size, we are also concerned with the brightness of the image. We can see that the amount of light entering the camera is proportional to the area of the lens. If D is the diameter of the lens, then its area is <^D2/4. This means that the image brightness is proportional to D2. The brightness of the image also depends on the image size. The more the image is spread out, the less light reaches any small area of the film or detector. The linear image size is proportional to f, so the image area is proportional to f2. This means that the image brightness is proportional to 1/f2.

Combining these two results, we find that the image brightness is proportional to (D/f)2. The quantity f/D is called the focal ratio, so the brightness is proportional to (1/focal ratio)2. We adjust the focal ratio in a camera by changing f-stops. Since the focal length of the lens is fixed, we change the focal ratio by changing the diameter of a diaphragm that controls the fraction of the total lens diameter that is actually used. Each f-stop corresponds to a factor of ^2 in the focal ratio, meaning that the image brightness changes by a factor of 2.

The discussions so far on image formation are really only appropriate for thin lenses, as well as optical systems where all of the angles are small. In real optical systems, rays that enter parallel do not all leave parallel to each other. Imperfections in the images formed by optical systems are called aberrations. Some of the aberrations are reduced by using the central part. The less we use the edges of the lens the better the images. That is why we might choose to use a diaphragm in a camera to block out the outer part of the lens. In a real optical system there is a tradeoff between image brightness and image quality.

One type of aberration is called spherical aberration. It arises from the fact that spherical curves are the easiest to grind on glass surfaces. These spherical shapes are close to the shapes required for proper image formation, but differ slightly, so the images are imperfect. Another type of aberration is called astigmatism. It occurs when the focal length depends on where around the lens the light strikes.

One aberration that occurs in lenses but not in mirrors is called chromatic aberration (Fig. 4.4).

9 Chromatic aberration.The focal length is different for different wavelengths.

This happens because a material's index of refraction depends on the wavelength. The focal length of a lens is therefore different at different wavelengths. The images at different wavelengths are formed in different places. We can correct, somewhat, for chromatic aberration with a two-lens system, called an achromat. The two lenses are made of different materials, with different indices of refraction, and different variations in the indices of refraction with wavelength. An achromat only brings the images at two wavelengths together, but images for intermediate wavelengths are not far off.

Now that we have seen some of the basics of optical systems, we can look at astronomical telescopes. Most current astronomical research is done on reflecting telescopes. However, the basic ideas of image formation in reflecting and refracting telescopes are the same. It is easier to visualize refracting telescopes so we consider them first.

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