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# 9.3.5: Applications

## Cameras

Here we’ll model a camera as a hollow box with film on the back wall, which acts as our screen, where our (real) image should ideally be located at. The lens has a fixed focal length $$f$$, and is able to slide in and out of a tube in front of our box. While this is a rather simple model, it is sufficient to explain how most cameras (both film and video) work.

Note that since the camera lens produces a real image, it will appear upside-down on the film negative. This is taken into account in the film developing and printing process; this negative is used to project yet another real image onto the photograph print. This print is then right-side-up, unless your film developing service loads the negative into their processor incorrectly. If you have a Polaroid camera, which makes a direct print from exposure, the light from the lens must be flipped off of an internal mirror before exposing the Polaroid picture.

Since our simple camera has a lens of a fixed (positive) focal length $$f$$, then the lens to image distance $$i$$ must vary for different object distances $$o$$. In fact, if you inspect the thin lens equation, as the object distance $$o$$ decreases (since $$f$$ is fixed), then the image distance $$i$$ must increase. You may have seen this for yourselves, as the lens barrel on your camera must be moved outwards to focus on close-up objects.

Exercise

Why can’t you make a camera that focuses light onto a film with a diverging lens?

# The Eye

In contrast to a camera we cannot change the image distance significantly for our eye. That is because the lenses are at the front of the eye (both a crystalline lens and the cornea contribute to the bending of rays) and the receptors are on the retina, located at the back of the eye. To get a clear image the image distance $$i$$ must be the same as the diameter of our eyeball. For most people this distance is roughly 1.71 cm. Because both the eye and a camera require focussing light onto a screen, they both require converging lenses.

### Accommodation

So if we cannot change $$i$$ why can we see things at a variety of distances? Unlike camera lenses, the lenses in our eyes can change focal length. We recall that a lens works by refraction, and while we cannot change the refractive index of our eye, the muscles around the eye, referred to as the ciliary muscles, can distort the lenses' shape.

When the ciliary muscles are relaxed the lens is (relatively) flat, and the light rays are not bent much as they pass through. This results in a larger focal length, because it would take a long distance for light rays parallel to the optical axis to converge to a point. When the ciliary muscles contract, the lens becomes more round, and the normals change more. This corresponds to more bending of the light as it passes through the lens; the rounder lens has a smaller focal length. Our ability to change the focal length of our eyes is referred to as accommodation.

### Near Point

There is a limit to how much the lenses in your eyes can change shape. Consequently there is a shortest focal length $$f_{min}$$ that your eyes can have, and a closest object that you can focus on clearly. The nearest distance that you can hold an object while still clearly focussing on it is called your near point $$d_{np}$$. Note that although it is called a "point" we actually refer to a distance. It is not the same as the shortest focal length, $$f_{min}$$ rather they are related by $\dfrac{1}{f_{min}} = \dfrac{1}{d_{np}} + \dfrac{1}{i}$

In practice it is much easier to measure $$d_{np}$$ than $$f_{min}$$, because to measure $$d_{np}$$ you only need to measure how close you can bring an object to your eye while still being able to focus on it. The nominal value for the near point of a middle-aged person is around 25 cm.

### Far Point

Similarly there is a furthest distance you can focus on when you totally relax your eyes. This distance is known as your far point, $$d_{fp}$$. For “normal” eyesight the far point is infinity – there is no furthest distance someone can focus on. However, if your eyes cannot relax completely, or your relaxed focal length is longer than your eyeball’s diameter, then you will have a far point. That is, you will not be able to focus on objects beyond some distance $$d_{fp}$$.

### Common Eyesight Defects

Let’s now consider three common defects of eyesight. Presbyopia (literally, “elderly eyes”) is nothing more than the normal loss of accommodation with advancing age. Children can read books much closer to their face than adults, because their near points are very short and their eyes are able to accommodate quite strongly. This ability decreases with age, so near points for children start to lengthen from as close as 10 cm, out to 25 cm by middle age (the nominal value for the near point), to even arm’s length or longer for older people. Typically, everyone will eventually develop presbyopia. When a presbyopic person’s near point is farther than $$o = 25\text{ cm}$$, glasses or contacts are prescribed to correct this vision defect.

Farsightedness (or hyperopia) is a condition where only far objects can be seen clearly. This is either because the lens cannot become round enough, or because the distance between the retina and the lens is too short. A farsighted person can see distant objects just fine with slight accomodation. As objects get closer, the eye must accomodate more strongly to focus images onto the retina. After a certain point, the eye cannot accommodate any further, and near objects remain out of focus.

Typically, children with hyperopic eyes will not have a problem with their vision, because they can strongly accommodate their eyes so they can see objects at any distance. However, as they gradually lose that ability as they grow up, then they will gradually not be able to see close-up objects. When a hyperopic person’s near point is farther than $$o = 25\text{ cm}$$, then glasses or contacts are prescribed to correct this vision defect. This is somewhat similar to presbyopia, but since accommodation is needed to focus on objects at all distances in hyperopia, eventually hyperopic people will need glasses (i.e., bifocals, or even trifocals) in order to see all object distances when accommodation is lost.

Nearsightedness (or myopia) is the condition where only nearby things can be seen clearly. This is because the relaxed lens is too curved, or that the retina to lens distance is too long. Since the ciliary muscles can’t “unaccommodate” a lens and flatten it out, there is no way that a myopic eye can see distant objects. As a myopic person’s far point is closer than $$o = \infty$$, then glasses or contacts are prescribed to correct this vision defect. However, while it is still relaxed, a myopic lens is able to focus on midrange objects. Accommodation easily allows the lens to focus on nearby objects.

## Corrective Lenses

When considering corrective lenses, we only need to worry about whether or not a (clear) final image can be made on the back of the retina. The way we go about this is by recalling that the eye itself is a lens, and responds the same to light that comes off an object directly and light that appears to be coming from the image of some object (as would be the case when we are wearing corrective lenses).

This gives us a strategy for modelling corrective lenses. We need to use a corrective lens because the object that we wish to focus on is closer than our near point or further away than our far point. The corrective lens creates an image of the object, and as we learned in our treatment of multiple lenses looking at the object through the corrective lenses is indistinguishable from trying to use our uncorrected eyesight to look at the image of the corrective lens. Provided the corrective lens places the image between our near point and our far point we will be able to see the object in question. By giving an object range that we wish to see (e.g. all objects up to 10 cm from my face) and knowing the near and far points, we can figure out what the focal length of the corrective lenses is required.

Typically when people quote the “strength” of lenses the number quoted is not the focal length. Instead it is the number of optical strength, which has SI units of diopters (D) $1\text{ D} = \left( \dfrac{1\text{ m}}{f} \right) \text{ m}^{-1}$

Note this means that the less lens correction needed (which corresponds to less bending, and a higher focal length) corresponds to a lower number of diopters. Also note that depending on whether converging ($$f > 0$$) or diverging ($$f < 0$$) lenses are needed, the prescriptions for the lenses can be either a positive or negative number of diopters.