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FROM LENSES TO OPTICAL INSTRUMENTS

by Giorgio Carboni
English version revised by David W. Walker

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CONTENTS

 


INTRODUCTION

The role played by optical instruments in the advent of the contemporary age is seldom taken into account. However the telescope has been decisive in the affirmation of the Copernican universe claimed by Galileo. Without the evidence yielded by this instrument during the observation of celestial bodies, the clash between the geocentric and heliocentric conceptions might have continued endlessly. Also the microscope performed a similar revolutionary function in biology and medicine, opening immense horizons for them. Before the appearance of the camera, the world descriptions were produced by artists only. They were valuable representations in which, however, the artist's subjectivity modified the reality. The camera introduced a much harsher way of observing the world, but much more objective.

Already from these short considerations, it is possible to guess how great the role played by optical instruments was in the formation of the world we know. These instruments are perfectly suitable to the modern and objective way of observing reality, but how do they work? Today we are in continuous contact with optical instruments and with their products such as pictures. Understanding the properties of lenses is fundamental to becoming familiar with these instruments, to use them with confidence and to use lenses in a creative manner in order to design optical instruments. This is exactly what we shall do in following articles and for that it is necessary to have a basic knowledge of optics.
Starting from the complicated theoretical descriptions in a physics book, the understanding of lens properties is not easy. However, by means of some simple experiments instead, it is possible to overcome many abstract obstacles. At this point, the return to the physics text will be simpler and more effective.

There are two types of lenses: the convergent ones and the divergent ones. The convergent lenses are able to converge the light of the Sun until to form a little and very bright disk that is the image of our star, while the divergent lenses diverge the bundle of light coming from the Sun so its image cannot form. Here we shall deal only with convergent ones, which are more important. The first experiments we shall perform now are intended to show the main properties of convergent lenses. The last one, through the combined use of two lenses, will show how some important optical instruments such as the telescope and compound microscope work.
A converging lens can be used in two main ways: as an image producer and as a magnifier.


THE LENS WHICH PRODUCES IMAGES

Equipment: a convergent lens with focal length between 100 and 300 mm, a candle, a white box, a meter rule. Buy the lens in an optical or photographic shop. Clear a table and prepare an "optical bench" like the one showed in figure 1. The p and q distances must be greater than the lens focal length. Light the candle and switch off the light. Modify the p and q values, until the candle image appears distinct on the box which you are using as a screen. Perform multiple tests, changing the distances. Try also exchanging the p and q distances.

How is the image formed? In order to explain this, normally two fundamental properties of lenses are taken into account:
- deviating a light beam parallel to its own axis, then making it to pass through the focus;
- leaving unaltered the path of the rays which pass through the lens center.

 

With reference to figure 2, take into account any object point, for convenience the extreme one. Among all light rays starting from this point, there are 3 whose path is particularly easy to follow. The A ray passing through the lens center and which is not deflected; the B ray which comes to the lens moving parallel to the axis and which passes through F1; the C ray which in a similar way passes through F2 and leaves the lens parallel to the optical axis. These three rays form an image point where they cross one another.

Operating in the same way for the other object points, you obtain the whole image. To trace these schemes, only two of these rays are required. There are also other rays, not parallel to the axis and not passing through the focuses, which contribute to the image formation. Also for these it would be possible to calculate the ray path, but to describe how a lens works, the ones we have taken into account are sufficient.

During this experiment, you will see that the image formed is inverted. This can be easily explained following the A ray path. In fact a ray starting from a high position on the object, after passing through the lens center, will be inverted on the image side.


MAGNIFICATION OF A LENS PRODUCING IMAGES

While performing experiments like those described in the previous paragraph, measure the height of the object and that of its image (fig. 3). Since the candle flame does not have a stable image, replace the candle with an object well lit by a lamp as shown in figure 5. If it is necessary, mask any stray light which does not pass through the lens, so as to obtain a higher contrast and a more visible image.

The size of the image is not invariable, in fact as the lens is moved towards the object, the image moves out and becomes larger (therefore you must move the screen away). The magnification is given by I=H/h, where H is the height of the image and h the one of the object.

It is not always possible to measure those dimensions. For example we cannot open a camera with the film inside, to measure the image. It is difficult even to measure very distant or too small an object. In these cases, the magnification can be determined by measuring the distances p and q. In fact, for thin lenses, the ray passing through the lens center and which is not deflected (fig. 3), contribute to forming two similar triangles which have a common vertex at the lens center. On the basis of the properties of similar triangles H/h=q/p, and, since I=H/h, also I=q/p. Verify this relation experimentally.

As you bring the the lens towards the illuminated object, you come to a position in which the image is far away. If the lens object distance is equal to the focal length, the image will be formed at infinity, whereas an object placed at infinity will produce its own image at the focal point. Furthermore, a lens placed at 2F from the object, will form the image at the same 2F distance. In this case, the magnification ratio is equal to 1.


DETERMINATION OF THE FOCAL LENGTH

What is the focal length? This word comes from the Latin "focus" (fire) for the lens' property of concentrating the sunlight so much as to set fire to combustible objects. The distance from the lens at which those objects must be kept has been named focal length. In optics this word is defined as the distance from the lens node (we will see that later) to the point at which a ray, which was initially parallel to the optical axis, intercepts the axis after being deflected by the lens.
To determine the focal length of a converging thin lens, use again your special optical bench. Arrange the illuminated object and the lens in such a way as to obtain a sharp image on the screen. Measure the p and q distances with the meter rule. The focal length is given by:

1     1     1                            p x q
--- = --- + ---  in explicit form:  F = -------
F     p     q                            p + q

To obtain a better approximation, more measurements must be made to calculate the average value of the lens focal length.


FOCAL LENGTH AND DIOPTERS

There is another way to indicate the focal length of a lens. In the fields of the production and the market of eyeglasses, instead of focal length people prefer speak of lens power, measured in diopters. So, if you have to buy an eyeglass lens, you need to know its power. Focal length and power of a lens are bound to each other and you can easily pass from one to the other using this simple formula:

D = 1/FL

where:
D = diopters
FL = lens focal length (expressed in meters!)
Besides, people place the sign "+" before the power of a converging lens and the sign "-" before the power of a diverging lens.

Let's make a couple of examples:
- a converging lens of half a meter of focal length has a power of +2 diopters. In fact: D = 1/0,5 = +2
- a diverging lens of 4 meter of focal length has a power of -0.25 diopters. In fact: D = 1/-4 = -0,25


THE MAGNIFYING LENS

Equipment: a convergent lens with focal length included between 20 and 60 mm.
1) Observe with the naked eye an object placed at a distance of 250 mm;
2) observe the same object with the lens and compare the two images.
The lens must be kept close to the eye. If it is planoconvex, keep the plane surface towards the eye. Approach the object until it becomes distinct.

This experiment is very simple. But how does the lens magnify the object?
The nearest distance of distinct vision with the naked eye is considered to be 250 mm. A normal adult man has difficulty seeing clearly objects closer than 250 mm. Converging lenses allows us to approach the object well below this distance and to still see it clearly. As we approach the object we will see it larger (fig. 4). A human eye is able to work with parallel light (from distant objects) or with light of limited divergence (objects not nearer than 250 mm). Converging lenses reduce the divergence of rays coming us from an object nearer than 250 mm, and allows us to still see it clearly.

The object to be observed must be placed between the front focus (F2) and the lens (fig. 4). For convenience we assume that the optical center of the eye coincides with the back focus (F1) of the lens. (The distance of the eye from the lens is not important, but in practice we will keep the eye close to the lens). Let's consider an object point. Among all the rays leaving the object we shall take for convenience ray A parallel to the axis, which is deflected by the lens and passes through the back focus F1 and arrives at the retina. We shall also take ray B passing through the lens center which is not deflected, and enters the eye where it is deflected by the cornea and intercepts ray A on the retina, forming an image point. The image formed on the retina is seen in a plane conventionally placed at a distance of 250 mm from the eye. It is not a real image, in the sense that it cannot be recorded on film and for this reason it is called virtual.
This image is perceived the right way up, although in the eye it is upside-down. Even when we are not using lenses, the images formed in the eye are inverted. It is the brain that corrects this image.

At the onset the A and B rays have a great divergence; on the other side of the lens, their divergence is reduced. If the object were placed in F2, the lens would make the A and B rays parallel, and to see the image clearly, the eye would focus at infinity. Finally, as we were saying, the magnifying lens reduces the divergence of the light coming from a close object. The lens also allows the object to be viewed clearly and magnified even below 250 mm.
Notice that the same converging lens can be used both as a magnifier and as an image generator. Note that the lens producing images turns them upside-down, while the magnifying glass keeps them the correct way up.

In the case of the magnifying lens, the magnification power is determined by the following relation: I=250/F, where F is the lens focal length (mm) and 250 is the conventional distance for distinct vision or reading. For example, a lens with 50 mm focal length will magnify 5 times. This is valuable when the eye is focused at infinity, whereas when it is focused for near vision, the relation becomes I=(250/F)+1. Hence, the lens of 50 mm focal length magnifies from 5 to 6 times, according to the eye's accommodation.
In a previous article, in which we talked about a little glass-sphere microscope, you could see to what extent a lens can magnify. However it is necessary to say that this is an extreme situation: normally a magnifying glass does not exceed 20 X.


THE OPTICAL INSTRUMENTS

Now you are finally ready for the conclusive experiment, the one that should enable you to understand how some of the most important optical instruments work. Let's go back to the optical bench. However, this time replace the box with a translucent screen. You can make it with a card frame on which you have fixed a piece of white plastic taken from a plastic bag (fig. 5). Focus the image on the screen and you can observe the image appearing from behind the screen.

You can also enlarge the image with a magnifying glass by taking the lens you used for the previous experiment and observing the image behind the translucent screen. As you can see, the image appears magnified. So far there is nothing strange. While you continue to watch the upside- down image, try to move the screen a little. The image keeps steady. Oh, dear! Then...
Remove the screen. Miracle! The image stays there. It is "floating" in the space. Therefore the screen was useless! It actually was! Not only is the image clearer and brighter, it is colored and in 3D too.

There, you have built a telescope! The lens nearest the object is your objective, the one near the eye is the eyepiece (fig.6).

 

Continuing this experiment, if the objective is brought closer to the object, the image moves away and becomes larger. Regulate the p and q distances in such a way that the image becomes larger than the object. Observe it with the magnifying lens: in this way you have obtained a compound microscope (fig. 7).

So, what distinguishes a microscope from a telescope? As you can see, the optical structure is the same, but with telescopes objects are distant whereas in a microscope they are close. Normally a telescope observes objects typically placed at hundreds of meters or more, and a microscope observes objects placed at a few millimeters or less from the objective.


REAL OR THICK LENSES

Up to now, we have dealt with thin lenses. This last paragraph introduces the concept of nodes and gives you an idea of the type of errors which occur when thin lens formulas are applied to real lenses. Thin lenses are considered to have no thickness, hence it is considered that the ray paths deviate when they meet the plane of the lens (fig. 8/A). In reality, the ray path is rectilinear inside an homogenous medium, and deflects when entering another medium with a different refractive index. Therefore, a ray passing through a lens is bent when it enters the glass and bent again when it leaves the lens (fig. 8/B).

In order to introduce the concept of a node, let's consider the ray which, among all those entering a real lens, does not deviate as it passes through the lens (fig. 8/C), and links the entering and leaving point of incident and emerging rays. Extending the path of both external rays to their intersection with the optical axis, locates two point which are named nodes. The focal length of a lens is referred to as the closer node.

The nodes of real lenses and their distance are neglected by thin lens formulas and this leads to some errors. However, for first drawing an optical instrument, or whenever one does not need high accuracy, this error is small and can be accepted in exchange for simpler calculations. In fact, the formulas for thin lenses are widely used by opticians for preliminary calculations and analysis, and are of value in most cases. But, especially if you are dealing with thick lenses of short focal length, or if you need precision, you must refer to formulas for thick lenses which you can find in optics texts.

CONCLUSIONS

The converging lens can:
- produce a real image of an object;
- magnify the apparent dimensions of an object or an image;
- be used with other lenses to build optical instruments.

I hope that these simple experiments have been able to introduce you to the world of optics. Now what do you need to make a telescope or a microscope? Just a bit of spirit of adventure. If the first instruments will be made up with "bad" lenses, no matter, on the contrary it is an important step to understand why in these instruments the objectives and the eyepieces are formed by more than one lens, but this is another story.


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In a similar form, this article has been published in the magazine "Scienza & Vita" of February 1994.


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