Magnifying glasses and telescopes are used to make things look bigger. In this chapter we will look at how they do this, and how much bigger an object looks when using a magnifier or telescope.
The apparent size of any object is related to the size of the retinal image it produces. All things being equal, an object whose retinal image is twice as big as the retinal image of another object will look twice as big 1. However, the retinal image size is quite difficult to measure directly, so instead we use the visual angle of an object as a surrogate for the size of the retinal image. The visual angle of an object is the angle between two imaginary lines drawn from the eye to the top and bottom of the object Figure 1. The retinal image size is proportional to the visual angle.
A magnifier or telescope makes an object look bigger by increasing its visual angle. The angular magnification produced by an optical device (like a telescope) says how much the visual angle is increased by the device,
(This is different from linear magnification, which is the ratio of the image height to the object height.) When the visual angles are small (which is often the case), the mathematics can be made a bit simpler by using the tangent of the angle instead of the angle itself. (For small angles, the tangent of an angle and the angle are proportional. The tangent of a visual angle is the height of the object divided by its distance from the eye.) Thus, for small angles, angular magnification can be defined as
A magnifying lens, or magnifier, is just a strong positive lens that creates a virtual image of the object being magnified. When a positive lens creates a virtual image, it is both larger and further away than the object. Because the virtual image is larger, this leads to magnification ( Figure 2 (a) to (c)).
To work out the amount of magnification, we first need to put in some distances. These are shown in Figure 3(a). In this figure, \( h_{obj} \) and \( h_{img} \) are the height of the object and virtual image respectively, and \( d \) is the distance from the lens to the eye (this is positive). Figure 3(b) shows \( d_{obj} \) , the distance from lens to object (negative), and Figure 3(c) shows \( d_{img} \), the distance from lens to virtual image (also negative).
Note that \( d_{obj} \) is negative.
Note that \( d_{img} \) is negative.
Using the facts presented in Figure 3, we can work out the magnification produced by the lens as follows:
| Step | Reason | Statement |
|---|---|---|
| 1 | Tangent of visual angle of object from Figure 3(b) | \( \tan(\text{visual angle of object}) = \dfrac{h_{obj}}{d-d_{obj}} \) |
| 2 | Tangent of visual angle of virtual image from Figure 3(c) | \( \tan(\text{visual angle of image}) = \dfrac{h_{img}}{d-d_{img}} \) |
| 3 | Magnification \( M \) is ratio of statement (2) and (1) | \[ M = \dfrac{\left(\dfrac{h_{img}}{d-d_{img}}\right)}{\left(\dfrac{h_{obj}}{d-d_{obj}}\right)}
\] |
| 4 | Factorize step (3) | \[ M = \dfrac{h_{img}}{h_{obj}}\times\dfrac{d-d_{obj}}{d-d_{img}}
\] |
| 5 | From Chapter 5, \( \dfrac{h_{img}}{h_{obj}}=\dfrac{d_{img}}{d_{obj}} \) | \[ M = \dfrac{d_{img}}{d_{obj}}\times\dfrac{d-d_{obj}}{d-d_{img}}
\] |
This formula for magnification is not all that simple, unfortunately. We can make it simpler by assuming that the object is at the focal point of the lens (that is, \( d_{obj}=-1/F \), where \( F \) is the lens power). That ensures that we get a virtual image.
When \( d_{obj}=-1/F \), the image distance \( d_{img} \) becomes huge (infinite, in fact) and so \( d-d_{img} \) becomes essentially the same as \( -d_{img} \), because \( d_{img} \) is so much bigger than \( d \).
Then, continuing:
| Step | Reason | Statement |
|---|---|---|
| 6 | Substitute \( d-d_{img}= -d_{img} \) | \( M = \dfrac{d_{img}}{d_{obj}}\times\dfrac{d-d_{obj}}{-d_{img}} \) |
| 7 | Cancle \( d_{img} \) | \( M = \dfrac{1}{d_{obj}}\times\dfrac{d-d_{obj}}{-1} = \dfrac{d_{obj}-d}{d_{obj}} \) |
| 8 | Assume \( d_{obj}=-1/F \) | \( M = \dfrac{-1/F-d}{-1/F} \) |
| 9 | Multiply top and bottom by \( -F \) | \( M=\dfrac{1+dF}{1}=1+dF \) |
This formula for magnification suggests that you can get a very large magnification by by making \( d \) large i.e. holding the magnifying glass a long way from the eye. This is true, but not useful. If the distance from eye to magnifier \( d \) is large, you are also a long way from the object. If you're a long way from the object, the visual angle of the object without the magnifier is quite small, and so magnifying it a lot leaves you pretty much back where you started. In fact, as we will see in a later chapter, the visual angle after magnification is unaffected by the distance \( d \) between the magnifier and the eye.
Still, people like to know how much magnification they will get from a magnifier. Since people tend to hold magnifiers reasonably close, a typical distance is \( d=0.25\text{m} \), which leads to the “nominal” magnification of
This gives an idea of how much magnification you can expect from a magnifier of power \( F \), under normal usage. An even simpler magnification is to just use \( F/4 \), which is fine for most purposes, particularly when \( F \) is large.
Magnifying glasses are used to look at near objects, because the object has to be closer than the focal length of the lens. To magnify distant objects, we need a telescope. To understand how a telescope works, let’s begin with a single positive lens, which is called the objective lens of the telescope (because it's at the end of the telescope that points at the object). The power of the objective lens is \( F_{objective} \).
This objective lens forms an image of the distant object, as is shown in Figure 4(a). Because the object is very far away, the vergence of the light entering the objective lens is \( V_{in}=0 \), and so the floating image formed by the lens is at the focal point of the lens. (The distance from the lens to the image is thus \( 1/F_{objective} \))
The floating image formed by the objective lens will not be especially large. We can make it bigger by looking at it through a magnifier. The magnifier glass is placed so that the image from the first lens is at the focal point of the magnifier ( Figure 4(b)). The magnifier is called the eyepiece because it is close to the eye of the person using the telescope.
This sort of telescope is called an astronomical telescope. From Figure 4(c), the distance from the objective to the eyepiece (the telescope length) must be the sum of the focal lengths of the objective and eyepiece:
One other feature of this telescope is that if a bundle of parallel rays enters the objective lens, the bundle is parallel when it leaves the eyepiece. This is because parallel light entering the objective will be focussed to a point inside the telescope, and that point is one focal length from the eyepiece, so the light from that point will leave the eyepiece in parallel. This means that, in some sense, the telescope has a power of \( 0\text{D} \), because the vergence of the light isn’t changed as it passes through the telescope.
Figure 5 gives the information we need to work out the magnification of an astronomical telescope. In Figure 5(a) two parallel beams of light coming from a distant object have been picked out, one from the top of the object (reddish) and one from the bottom of the object (yellowish).The angle between the beams when they leave the telescope is greater than the angle between the beams when they enter the telescope, so the telescope has magnified the object.
Notice that the beam which was travelling downwards when it hit the objective is travelling up when it leaves the telescope, and the beam that was travelling up when it hit the objective is now travelling down. This crossover of the beams means that when we look through an astronomical telescope, the image we see is upside down.
In Figure 5(b), we have removed all but two rays. The two rays are ones that pass through the exact centre of the objective, and so by ray-tracing rule 3, they pass through the lens with their directions unchanged.
The magnification produced by the telescope is given by the two angles \( b \) and \( a \). \( b \) is the angle between the two rays before entering the telescope. The angle \( a \) is the angle between the rays after passing through the telescope. If you put your eye where the two rays crossed (to the right of the eyepiece), \( a \) would be the visual angle of the object with the telescope. If you removed the telescope and put your eye where the objective was, then angle \( b \) would the visual angle without the telescope.
However, we also have to think about the sign of \( b \). If we measure the angle \( b \) from the yellow ray towards the orange ray, we are measuring in a clockwise direction. This is usually considered a negative angle, so in fact we have \( \tan(b)=-h/d \).
If we again measure the angle from the yellow ray to the orange ray, we are now measuring in a counter-clockwise direction, which is usually considered positive.
The length \( d \) of the telescope is a positive number, so \( V_{in}=-1/d \) in this case.
The magnification of the telescope is then
However, there is one additional thing to worry about: angles have a sign convention too. If we look at the angle \( b \), it is the angle between the two rays. We might decide to measure the angle starting from the yellow ray and turning towards the orange ray. That is, we are measuring the angle in a clockwise direction. If we measure the angle \( a \) from the yellow ray towards the orange ray, we are turning in a counter-clockwise direction. So there are opposite directions.
Usually, when we measure an angle (and we care about the sign of the angle as here) we say that clockwise give negative angles, and counterclockwise turns give positive angles. That's what we will do here.
With that in mind, the magnification of the telescope can be worked out, with the help of Figures 5(c) to (e), as follows:
| Step | Reason | Statement |
|---|---|---|
| 1 | Tangent of visual angle without telescope (see Figure 5(c)): | \( \tan(b) = -\dfrac{h}{d} \) |
| 2 | Tangent of visual angle with telescope (see Figure 5(d)): | \( \tan(a) = \dfrac{h}{e} \) |
| 3 | Magnification = \( \dfrac{\tan(a)}{\tan(b)} \) | \( M = \dfrac{(h/e)}{-(h/d)} = -d\times\dfrac{1}{e} \) |
| 4 | Thin lens equation (see Figure 5(e)) | \( \dfrac{1}{-d}+F_{eyepiece}=\dfrac{1}{e} \) |
| 5 | Substitute \( 1/e \) from Step (4) into step (3) | \( M = -d\left(\dfrac{1}{-d}+F_{eyepiece}\right) \) |
| 6 | Multiply out | \( M = 1-d\times F_{eyepiece} \) |
This magnification formula is already pretty simple, and shows that the magnification is given by the eyepiece power and the telescope length. Note that, apart from the minus sign, it is quite similar to the magnification formula for a magnifying glass.
However, we can change this formula to a different one which involves just the objective power and eyepiece power, as follows:
| Step | Reason | Statement |
|---|---|---|
| 7 | \( d \) is the length of telescope | \( d=\dfrac{1}{F_{objective}}+\dfrac{1}{F_{eyepiece}} \) |
| 8 | Substitute \( d \) from step (7) into (6) | \( M = 1-\left(\dfrac{1}{F_{objective}}+\dfrac{1}{F_{eyepiece}}\right)\times F_{eyepiece} \) |
| 9 | Multiply out | \( M = 1-\left(\dfrac{F_{eye}}{F_{objective}}+1\right) = -\dfrac{F_{eyepiece}}{F_{objective}} \) |
So the magnification of a telescope is simply the ratio of the lens powers
Note the minus sign in front of the fraction here, which means the view with the telescope is upside-down compared to the view without it. For example, a telescope with a \( +2\text{D} \) objective lens and a \( +6\text{D} \) eyepiece has a magnification of \( -(6/2)=-3 \). That means objects appear three times larger with the telescope than without, and the minus sign means they are upside-down.
The magnification of a telescope is increased when the eyepiece power is large or when the objective power is small. For example, the largest existing astronomical telescope using lenses (rather than mirrors) is the Yerkes telescope in the USA. The power of the objective in the Yerkes telescope is only \( +0.05\text{D} \) (which means that the telescope is over 20 metres long).
The astronomical telescope has two disadvantages: it tends to be long, and it inverts the image. The second disadvantage isn’t an issue if looking at objects in space, because there is no up or down there. This is why the telescope is called an astronomical telescope. However, the inversion is an issue when looking at objects on earth. The inversion can be fixed by inserting an extra lens to give the light rays one more flip, but this makes the telescope even longer. The 17th century astronomer Galileo invented a different telescope design which is shorter than the astronomical telescope, and which does not invert the image.
The idea behind it is this. The magnification formula for a telescope is
If we used a negative lens for the eyepiece, then the magnification would be positive, and that means the image won’t be upside down. For example, a \( +2\text{D} \) objective and a \( -10\text{D} \) eyepiece should give a magnification of \( M=-(-10)/(+2) = +5 \).
This is the basis of the Galilean telescope, shown in Figure 6 , which has a positive objective and a negative eyepiece. Just as with the astronomical telescope, the distance between the two lenses is the sum of their focal lengths. Since one of the focal lengths is negative, this makes the telescope shorter. As with the astronomical telescope, when parallel light enters the objective, parallel light leaves the eyepiece. But unlike the astronomical telescope, the image is not inverted.
The way the Galilean telescope magnifies is a little less obvious than the astronomical telescope. Look at Figure 7. As in Figure 5, two parallel beams of light are coming from the distant object, one from the top of the object and one from the bottom. We can selcted just two rays to look at ( Figure 7(b)), both of which pass through the optic centre of the objective lens. The two (divergent) rays then continue down the telescope until they hit the eyepiece. The eyepiece diverges them even more. The two rays leaving the eyepiece don’t cross, but we can extend them back as “virtual” rays until they do cross (( Figure 7(c))). The visual angle through the telescope \( b \) is the angle between the two virtual rays where they cross, and the visual angle without the telescope \( a \) is the angle when they hit the objective. Note that the ray which was travelling downwards when it entered the telescope is still travelling down when it leaves the telescope, and the ray which was travelling upwards is still travelling up. The rays aren’t crossed over inside the telescope, and so the image isn’t inverted.
Telescopes were designed to magnify distant objects, but they can be modified to magnify near objects. These modified telescopes are useful for low vision patients who require more magnification than is possible with magnifying glasses.
The problem with using telescopes on very near objects is that light from a near object is divergent when it hits the objective lens. Telescopes are designed under the assumption that the light entering the telescope is parallel. When divergent light from a near object enters the telescope, divergent light leaves the telescope (see top of Figure 8). An eye which has some accommodation may be able to focus this divergent light, but maybe not, so the divergent light leaving the telescope is a potential problem.
Fortunately, it's easy to fix. Since the telescope works best with parallel light, we need to turn the divergent light from the near object into parallel light before it enters the telescope. This can be done with an extra lens. For example, suppose we want to use a telescope to view an object at a distance of \( 8cm \). When we put the the telescope this far from the object, the light striking the objective lens has a divergence of \( -12.5\text{D} \). We can change this to parallel light by simply inserting an extra \( +12.5\text{D} \) lens in front of the telescope. The light leaving the extra lens is now parallel, and so the telescope works as expected. (see Figure 8, bottom). The magnification is calculated from the usual telescope magnification equation, ignoring the extra lens.
In a real low-vision telescope, one does not add an extra lens. Instead, one just changes the power of the objective, because powers of lenses add when they are put close together. For example, suppose we have a Galilean telescope with an objective power of \( F_{objective}=+2\text{D} \) and an eyepiece power of \( F_{eyepiece}=-10\text{D} \). We want to use this at a distance of \( 5\text{cm} \), so we need to put an extra close-up lens with power \( +20\text{D} \) in front of the objective. If the extra close-up lens is placed so there is no gap between it and the objective lens, the \( +20\text{D} \) lens and the \( +2\text{D} \) objective act the same as a single \( +22\text{D} \) lens. Thus we can built our telescope from a single \( +22\text{D} \) objective lens, which combines the telescope objective and the close-up lens into one, and a \( -10\text{D} \) eyepiece.
The extra close up lens does not change the magnification. One way of explaining this is to use the magnification formula from Step (6) earlier: \( M=1-d F_{eyepiece} \). The extra lens hasn't changed the length of the telescope, so the magnification hasn't changed.
The extra close-up lens (which is usually fused to the objective to form a single lens) allows you to use a telescope close to things and still get the desired magnification. If we use a very high powered close-up lens, then we can get very close to objects. In that case, we have a microscope.
Low-vision telescopes are often based on Galilean telescopes, because it is important not to invert the image. However, this doesn't matter for microscopes, so they are usually constructed as an astronomical telescope with an extra very powerful close-up lens (which of course is just fused to the objective lens to get a single powerful objective).