Cm

Figure 3.17 Least distance of distinct vision (not to scale).

When the muscles are relaxed, the power of the lens is P = 1/f = 1/0.02 = 50 diopters.

When one is looking at nearby objects, the tension of each ciliary muscle is increased. As a result the curvature of the surfaces increases and the focusing power increases. This process is called accommodation. At maximum power, an object at a distance of 25 cm can just be kept in focus, as illustrated in Figure 3.17.

We can calculate the maximum power easily from the lens equation:

0.02

A normal eye can increase its power from 50 to 54 diopters.

The image on the retina is inverted (see Section 3.3) — a fact which a baby realises when it has to reach up in order to touch something which must at first appear to be down below! In 1896, G.M. Stratton carried out an experiment, in which subjects had to wear special inverting glasses for an extended period. He found that, after a few days, the world which they saw through these glasses appeared to them to be quite normal. In fact, after they had taken off the glasses, they became confused and needed a further period of adjustment to get back to normal vision.

In front of the lens is the iris, within which there is a variable aperture called the pupil which opens and closes to adjust to changing light intensity. The receptors on the retina also have intensity adaptation mechanisms. As a result, the eye is adaptable to an enormous range of light intensity. For example, sunlight has an intensity 100,000 times greater than moonlight, which in turn is 20,000 times more intense than starlight. Yet one can easily distinguish objects at night, particularly when the moon is full.

3.6.2 Common eye defects

Myopia (short-sightedness)

Parallel light is focused at a point in front of the retina, as the power of the lens is too great. (Either the curvature of the lens is too large, or the eye is longer than is normal.) As a result the person cannot

focus clearly on points further away than a certain distance called the far point.

Diverging lenses will correct the defect.

Hypermetropia

(long-sightedness)

Even with maximum accommodation, the power of the lens is not sufficient to obtain an image of objects at a distance of 25 cm; the near point is further from the eye.

Such a person needs reading glasses with converging lenses.

Presbyopia

The inability to accommodate fully often increases with age. This defect may be corrected with bifocal lenses; the upper part of the lens is used for distant vision and the lower for close work.

Astigmatism

This defect is caused by irregularities in the lens, or elsewhere in the eye, and causes distorted vision. It can be compensated by using lenses with cylindrical or more complicated surfaces.

Whether we use contact lenses or ordinary spectacles, the separation of the glass lens from the eye is small in comparison with the object distance. We may make calculations on the basis of thin touching lenses.

The earliest spectacles were made in the 14th century. In 1363 Guy de Chauliac prescribed spectacles as a last remedy when salves and lotions had failed. Originally only convex lenses were used. Concave lenses were introduced later, in the 16th century.

3.7 Making visible what the eye cannot see

3.7.1 Distant objects

Telescopes enable us to see and examine objects which are far away; microscopes, to see and examine very small objects. The sun, the moon and the other planets in our solar system can be studied even with relatively simple telescopes. Most stars other than the sun are so far away that, even with the most powerful telescopes, we get a point image and it is just not possible to observe features that might be on the surface of the stars. However, a telescope does collect more light than the naked eye, so the image is very much brighter and countless new stars become visible.

Hubble stamp. Courtesy of An Post, Ireland.
Betelgeuse. Courtesy of A. Dupree (CFA), R. Gilliland (SIScI), NASA, ESA.

In 1995, NASA published a picture (taken with the Hubble Space Telescope) of the star Betelgeuse, consisting of a central disc with the halo of an atmosphere around it. Betelgeuse is one of our nearest stars, 'only' about 430 light years away, and is very large, with a radius about 1800 times the radius of the sun.

subtended angle 0.5 arcsec

- g : : = = : = = :-ft telescope Betelgeuse - 430 light years -

subtended angle 0.5 arcsec telescope

Figure 3.18 The angle subtended by one of our nearest stars.

Magnification, defined in terms of the ratio of the size of the image to the size of the object, does not of course make sense when we are viewing astronomical objects, particularly when we have a point image. However, when the image is of finite size, we can measure the power of a telescope in terms of the smallest angular size which can be resolved. This is the angle between rays from opposite sides of the object and is also a measure of the angle subtended by that object.

In the case of Betelgeuse, the angular size is about 0.5 arc seconds, equivalent to the angular size of a 50-cent coin at a distance of 50 km, as illustrated in Figure 3.18.

3.7.2 Nearer but not clearer

The situation is quite different in the case of an object at a finite distance from us. We can obviously increase the angular size of the object by standing closer to it. As we come nearer and nearer, we can distinguish more detail, but there is a limit. When we get inside the least distance of distinct vision, the clarity diminishes, and the image becomes blurred. Any advantage gained by increased angular size is lost.

'Inside' the least distance of distinct vision, the images of nearby points overlap, and the letters on the worm's chart merge. Things are nearer, but not clearer. The problem is that the rays of light from any point on the object are so divergent when they reach the eye that the lens is not strong enough to focus them on the retina. The 'task' is too great and the strain on the ciliary muscles will most probably lead to a headache!

We need a real image on the retina to transmit to the brain; rays from every point on the object should be sharply focused at a corresponding point on the retina if we are to see a 'true' image of the object.

As we get very close to an object, its angular size increases, but it becomes impossible to focus rays from any given point on the object.

Figure 3.19 shows two sets of rays, one emerging from the top of an object and the other from the bottom. The lens is not strong enough to focus either set of rays to a single point on the retina. The result is a jungle of overlapping rays reaching the retina and confused messages reaching the brain!

Figure 3.19 Too close for comfort.

Figure 3.19 Too close for comfort.

How can we increase the angular size of the object without making the rays from individual points on the object too divergent?

3.7.3 Angular magnification The simple magnifier

A single convergent lens may be used as a magnifying glass to provide the simplest method of increasing angular magnification without blurring. Placing the object inside the focal point of the lens gives a magnified virtual image, as shown in Figure 3.20. In fact, if we place the object at the focus of the magnifier, the rays from each point emerge parallel and can be viewed comfortably with the relaxed eye.

The angle subtended by the object as seen from the centre of the lens is: q @ tan Q = hf.

As far as the observer to the right of the lens is concerned, this is also the angle between the rays coming from points at the top and the bottom of the object. These rays are parallel and can be focused comfortably on the retina. The function of the magnifying glass has been effectively to bring the object close up to a distance f from the observer without blurring.

'Black' rays from point The angle between the upper and at infinity forming lower rays is 8 but all upper and lower top of the image. A rays are parallel and easy to focus.

'Black' rays from point The angle between the upper and at infinity forming lower rays is 8 but all upper and lower top of the image. A rays are parallel and easy to focus.

'Grey' rays from point at infinity forming foot of the image.

f = focal length of the magnifier

'Grey' rays from point at infinity forming foot of the image.

f = focal length of the magnifier

Figure 3.20 The simple magnifier.

The angular magnification is defined as the ratio of the angular size as seen through the magnifier (0) compared to angular size (p) when the object is placed as close as possible to the normal eye, at the least distance of distinct vision (d = 25 cm).

For small angles, angular size is equal to actual size/separation a h a h q « - and j « -f d fi angular magnification a ~ y

A normal good magnifying glass may have a focal length of 7.5 cm. The resultant angular magnification a ~ 25/7.5 ~ 3.3.

This means that effectively you can observe an object as if it were 7.5 cm from your eyes instead of 25 cm. It is difficult to make large convex lenses of greater power without encountering distortion due to non-paraxial rays, although small high quality lenses with focal lengths of the order of millimetres are made for optical instruments. To achieve greater magnifying power it is normally necessary to use a combination of lenses forming a compound microscope.

Antoon van Leeuwenhoek (1632-1723) was a brilliant lens-maker who hand-crafted lenses of high quality and very short focal length. No one was able to match his skill at the time and microscopy suffered a great setback after his death until the development of compound microscopes about 100 years later.

3.8 Combinations of lenses 3.8.1 Compound microscopes

The simplest compound microscope involves two lenses. The objective lens has a short focal length and the object under study is placed just outside the focal point of the lens. This produces a real, inverted image, which is much larger than the object. (The magnification is typically x 50.)

This image is viewed by the eyepiece, which is used as a magnifying glass. The final image is virtual, inverted and further magnified.

Figure 3.22 illustrates the principle of the compound microscope. The objective lens, with a focal length f1 (typically about

Figure 3.22 Compound microscope (not to scale).

3 mm), forms a real and inverted image of an object placed little more than 0.05 mm beyond its focal point, at a distance of about 15 cm. The magnification of this intermediate image is m « -v/u ~ 15/0.3 = 50 . If the object has a lateral size of h, the size of the intermediate image is 50 h. This image is placed at the focus of the eyepiece and forms an object for the eyepiece. The angular magnification, a ~ d/f2 , where f2 is the focal length of the eyepiece and d is the least distance of distinct vision.

Angular magnification of the final image = 50 x 25h = 312.5.

The notion of an image 'at infinity' suggests that such a thing might be difficult to examine. In fact, the opposite is true. The relaxed eye can comfortably focus parallel rays on the retina. The final image produced by the microscope serves as an object for the eye and can be imagined as being composed of a series of points. Rays reaching the eye from any one point are parallel. Rays from different points are not parallel. In fact, the angle between the beam of rays coming from the point at the top and the beam from the point at the bottom determines the angular magnification.

In high power microscopes, the eyepiece and objective lenses are themselves combinations of lenses. Such combinations are designed not only to increase magnification but also to minimize distortion due to lens aberrations. In addition, all but the simplest microscopes have binocular eyepieces, essential if they are to be used for an extended period of time.

When one is looking at an object close at hand, it is unnatural to focus one's eyes for infinity and even microscope operators may take time to adapt. It may also take time to adapt to binocular eyepieces so that one image is seen by both eyes, and not two. In cases where it matters, one must also get used to the fact that the image is upside down!

3.8.2 Telescopes

The Dutch optician Hans Lippershey (1570-1619) succeeded in 'making distant objects appear nearer but inverted' by means of a tube inside which were mounted two spectacle lenses. He applied to patent his invention, but it was refused on the basis that the instrument would be 'too easy to copy'.

The principle of the telescope is similar to that of the microscope, in that the objective lens forms an image, which serves as an object for the eyepiece and produces a final image which is inverted. The main difference between the two is that the purpose of the telescope is to look at distant objects, while the microscope is designed to look at small objects close at hand.

In the refracting telescope the objective is large, to gather as much light as possible, and has a relatively large focal length.

In Figure 3.23 the eyepiece is a simple convex lens. The incoming light passes through the focus of the eyepiece, which forms an inverted final image.

In the Galilean telescope (Figure 3.24), the objective is a convex lens and the eyepiece is a concave (diverging) lens or lens system.

A concave lens placed inside the focal point F of the objective acts in the same way as a convex lens placed at or beyond F, objective lens objective lens

Figure 3.23 Refracting telescope.
Figure 3.24 Galilean telescope (the image is upright).

except that the image produced by the concave lens is not inverted. This has an obvious advantage for terrestrial use.

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