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But these tools do not belong in the Introduction. These techniques are what this book is about. As we saw in the introduction, for many years this was the best microscope.

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Simple microscopes, as single-lens microscopes are called, can give high-resolution images. Once we understand how one lens works, it is easy to see how lenses can be combined to form both ancient and modern compound microscopes. The distance from the lens to the focus is called the focal length. For a simple understanding of the microscope, the focal length is the only thing we need to know about a lens. This simplification ignores any imperfections of real lenses, assuming that all lenses are perfect, but it is a great starting point.

We can use the focal length to draw some very simple ray diagrams that will show us how and where a lens forms an image, and from there, how different lenses work together in the complete microscope. Because all parallel rays are deflected through the focus, while rays that pass through the exact center of the lens go straight through, undeflected, it is easy to plot where the image of a specimen appears. Provided that the object is farther away from the lens than its focal length, we can draw these two rays, as shown in Figure 1.

Where the rays meet is where the image of that point will be. If the sample is closer to the lens than its focal length, the two rays will not meet. Figure 1. In fact, you can easily project an image in this way if you take out an eyepiece from a microscope in a darkened room. Put a piece of tracing paper over the hole, and you will see the image, because it is formed at a point near the top of the microscope tube. A slight adjustment of the fine focus should bring the image into sharp focus.

How, then, can we get a greater magnification? As Figure 1. A so-called higher-power lens really means a lens with a shorter focal length. To form the image at the same place as the low power lens, we must bring the specimen closer to the lens; this is why highpower objectives are always longer than low-power ones. Objectives are designed to be parfocal: As you rotate the turret of the microscope, each lens should come into approximate focus once one has been focused. In theory and indeed in practice we could make the image larger with any given focal-length lens by forming the image farther away i.

It forms a more highly magnified image, but at the same place. Having a fixed tube length brings us the added benefit of knowing the magnification an objective of a given focal length will give, and we can mark this on the lens. If you look at an old microscope objective, from the days when tube lengths were not standardized, you will find that the objective gives the focal length rather than the magnification. Usually, we do not view the image on a sheet of tracing paper, but through another lens, the eyepiece.

The eyepiece is positioned so that the real image projected by the objective is closer to the lens than its focal length.

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This means, as we have seen, that the eyepiece cannot form a real image. However, if we again plot the rays using the focal length Figure 1. The virtual image has the same orientation as the original real image but is much larger, so the final image remains inverted relative to the specimen but is magnified further by the eyepiece. It cannot therefore form a real image; instead, the rays appear to originate from a magnified virtual image. As we adjust the focus, we control where the image forms. For the sake of simplicity, an old single-eyepiece microscope is used, because its optical path is in a straight line.

Note that microscope resolution has not improved since the end of the 19th century; the modern microscope is not really any better optically. Each operator may focus the microscope slightly differently, forming the virtual image at a different plane. By this time, the microscope had developed to the point where its resolution was limited only by the wavelength of light, so it has not improved since. Modern improvements are fundamentally matters of convenience Figure 1. A microscope fitted with a camera has crosshairs visible through the eyepiece, and the eyepiece has a focusing adjustment to bring the crosshairs into focus.

The microscope focus is also adjusted so that the image is seen in sharp focus with the crosshairs superimposed — making the image also in focus for the camera. The final magnification that we see will be the objective magnification multiplied by the eyepiece magnification, provided that the tube length is correct. The image captured by a camera will have a different value, determined by the camera adaptor. Modern microscopes rarely have just one eyepiece, a design that sufficed until the midth century. Binocular eyepieces use a beamsplitter to duplicate the same image for both eyes; they do not give a stereoscopic view Figure 1.

In this, they differ from the beautiful two-eyepiece brass microscopes of the 19th century, which were stereoscopic but sacrificed some image quality by looking through the objective at an angle.

Different microscope operators must always make an adjustment for the interocular spacing to suit different faces, and they must do so without altering the tube length. Sometimes, a pivoting arrangement exists so that the actual length does not change. On other microscopes, the eyepieces move out as they come closer together and move in as they move farther apart.

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Some older microscopes must be adjusted manually; the focusing ring on each eyepiece must be adjusted to match the scale reading on the interocular adjustment. If we construct ray paths through a lens as before, but for two points on the sample, equidistant from the optic axis, we get Figure 1. Both eyes see the same image.

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Rays at corresponding angles cross at the same plane as the focus the back focal plane, BFP but off the optic axis. With these two rays from each point, we have defined the position and size of the image.

Because we know that all rays from each point must arrive at the corresponding point on the image, we can now draw in as many rays as we like. An interesting feature of the ray paths is now apparent.

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You may like to plot rays at larger angles and observe where they meet — plotting it yourself explains more than any number of words. The construction lines have been omitted for the sake of clarity; in any case, having already defined two points on our image, we know where other object points will be imaged. These constructions show that at the plane of the focus, all rays, from any point of the object, which leave in one particular angle and direction, meet at one point. The larger the angle, the farther this point is from the optic axis.

This plane is called the back focal plane BFP, or sometimes just the focal plane of the lens. It is also sometimes called the pupil. The back focal plane of the objective is easy enough to see. Simply focus the microscope on a specimen, and then remove an eyepiece and look down the tube.

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You can obtain a larger and clearer image by using a phase telescope instead of the standard eyepiece. This is a long-focal-length eyepiece, which takes the back focal plane as its object and gives a magnified virtual image of it, just as a standard eyepiece does with the image. Alternatively, some of the larger and more elaborate research microscopes have a Bertrand lens built into the column Figure 1.

This is an extra lens, which can be put into position with a slider and forms an image of the back focal plane at the normal image position, where it is magnified by the normal eyepiece just as if it were the image of the specimen. Although the built-in illuminator and binocular eyepieces make it much more convenient to use than its older counterpart in Figure 1.

The modern microscope focuses by moving the stage, whereas the older one moves the tube carrying objectives and eyepiece. The arrow indicates the Bertrand lens, which permits the BFP to be viewed through the eyepieces. Because the sample has a regular pattern, it scatters light in particular directions, and this pattern is visible at the BFP, where light is sorted by direction. The condenser must be closed down to give parallel illumination to see the pattern.

At the back focal plane, light can be sorted not according to where it comes from, but according to the angle at which it has been scattered by the specimen. This gives a handle on controlling and testing the resolution of the microscope; it also opens the door to a range of important contrast-enhancing techniques, which will be covered in a later chapter.

Magnification does not really answer the question — a huge image is no good if it is not sharp! Two brilliant 19thcentury scientists turned their attention to the problem and came up with rather different treatments. The controversy of X polarized the microscope community, and arguments raged as to which approach was right. The protagonists themselves — John William Strutt later Lord Rayleigh and Ernst Abbe — watched the debate in bewilderment, because it was clear enough to them that they were simply describing different imaging modes.