Introduction
Look at the picture of the Sun, on the right. Interestingly enough, this image was produced not using an optical telescope, but a collection of radio telescopes. A radio telescope is very similar to aTV radio dish antenna. It is just a detector. It collects radio waves (electromagnetic radiation with wavelengths in the range of a few centimeters to a few meters). In analogy to our own vision, each radio antenna plays the role of a rod or a cone. So, one such telescope could do little more than record the presence of a radio wave. But because the earth is revolving, even one detector could be useful in "getting a picture". All we would have to do would be to continue recording the signal over many hours and from these data create a "map", or a radio image. Of course the best way to create a full image would be to place many radio telescopes in a grid, the same as the collection of rods and cones in the eye. Then any snapshot could generate an image. This is in fact the way that the picture of the sun, shown on the right, was generated.
This image was created using the Very Large Array radio telescope system located in Sonora, New Mexico. This array consists of 27 separate dishes, each 25 m in diameter. These dishes (antennas) are movable on a Y-shaped rail track. (See the photo of the dishes and the array, to the right.) By adjusting the configuration of the array radio astronomers can create images that have a one-to-one correspondence with the source object.
One would be tempted to ask why only 27 antennas? Of course the cost of construction is an important issue. But as it turns out, as in the case of the eye, more is not necessarily better! In fact, in both of these situations more would be just wasteful! This is because of diffraction, a phenomenon particular to waves that we need to examine further here.
Diffraction
Diffraction is a very curious behavior of waves when they encounter an obstacle or an opening. When the size of the obstacle (or the opening) is large compared to the wavelength, nothing interesting happens. The obstacle just impedes the passage of the wave. For example, were you to place your hand in front of a light source, you could shield your eyes from the rays of light. This is exactly what we do when we are looking at a person or an object that is in front of a strong light source, say the sun. But when the size of the obstacle becomes comparable to the wavelength, then light actually bends about the obstacle. This bending of light, the diffraction, is a rather limiting feature of waves with regard to image forming. At one level, the limitation imposed by diffraction is responsible for the size of our rods and cones. At another level, it dictates how small of an object can be imaged by a given wavelength of light. These two issues are of course related, but they are still separate issues.
To examine the first of these issues remember that all we need to create an image is just a hole, a pupil. The smaller the pupil, the sharper the image. We saw this demonstrated in class. You may remember that as we used a smaller and smaller pupil size the image got sharper, but at the same time it got less and less bright. (Of course this also affected the field of focus.) At the time, however, we totally ignored the effects of diffraction. What if light intensity were not a concern? In the absence of diffraction, we could get as sharp of an image as we need just by making the pupil size smaller and smaller. But once the pupil size falls into the general neighborhood of the size of wavelength of light, then the diffraction effect will cause our image to get blurred. Consider the following two photographs. The one on the left is a photograph of two thin parallel wires (as thick as a human hair, but made of opaque copper metal) held in front of a cone of laser light. The one on the right shows two similar wires illuminated by the same light, but now the wires are crossed as in the letter X. Notice two interesting features. First of all, the cone of light (right photo) appears as a cone surrounded by another thin and hollow cone. So, its projection is seen as a circular disk surrounded by a ring. This in itself is due to diffraction and interference effects. We get this light pattern because the laser beam is sent through a very small pinhole (10 micrometers in diameter). So, the light bends around the circular aperture of the pinhole and creates the interference pattern that you see. Because of the interference there is the "dark" ring. So, it turns out that this pattern is not just a simple cone of light. However, for considering diffraction effect from the wires we can focus our attention to the central portion of the light, which is just a simple cone of light.
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Now, notice that the center of the two parallel wires is in fact bright! So, evidently, the light bends around the wires in such a way that instead of "blocking" the light by their presence, the two wires create light at the center of their "shadows"! Also, there are other shadows (bands) next to each image of the wire. Each light band is separated from the next light band by a dark band. Now, when the two wires are crossed, as shown on the right photo, there is a point at which we cannot tell where the shadows of the wires are. The light and dark bands seem to overlap so as to make it hard to see individual shadows.
For a single opening/obstacle of diameter a simple ray optics shows that at a distance D >> a successive mimima of diffraction occur at angular positions qm given by:
a sin( Qm ) = m l , where m=1,2,...
In the above l is the wavelength of light and the integer m denotes the "order of diffraction". So, for the very first minimum, m=1, next one m=2, etc. Alternatively, we could use the approximation:
sin( q ) = q = y/D, for D>> y
and write the lateral position of the minima, ym, in terms of the obstacle size a and wavelength l:
ym = m l D/ a
Lord Rayleigh examined a more detailed formulation of diffraction effects to show that for a circular openings of diameter a the first diffraction peak will have a minimum at
y = 1.22 l /a
He then assigned a criteria for resolving two such circular apertures to be that their central maxima should be separated by just the above mentioned distance.
What we are discussing, of course, is the issue of resolution. That is two say, how close could we place two thin wires and still recognize them as two separate wires? It is clear from these photographs that there is an "ultimate" resolution because of the diffraction of light. In the first issue, referred to above, this resolution limit set by diffraction tells us that there is a limit to the resolution of our eyes (or any imaging device) because of the aperture size of our eye's pupil (the circular light feature). So, it would be wasteful to have smaller cone and rod cells than we currently have! If the rods and cones were smaller, then our eye's camera-detector would have a resolution that would be better than the diffraction resolution of its pinhole. This would be wasteful. Likewise, the resolution limit depends on the wavelength of light, so the smaller the wavelength, the better the resolution it could provide. This is again through the consideration of the above photos. If we had used a light of shorter wavelength, then the diffraction patterns would have been denser and thus less effective in blurring the shadow images.
So, how could we improve on the resolution? Clearly, the answer is to use smaller wavelengths. How small is small? Visible light falls in the range of 700 to 400 nm. In optical microscopy we could use UV light of as low as 200 nm. The best this could do is to allow us to image objects that are about 200 nm in size. We could not see, for example, viruses, which are typically tens of nm in length. Is it possible to image smaller things? The answer is yes! (See pages on Imaging Molecules & Atoms.)
