Angles Of View
In a previous article we noted that the speed of light, "c," is the maximum velocity anything can achieve. Lying just beyond that observation is the fact that even light itself can only reach that fantastic speed if it is traveling in a vacuum. In other, denser media (like our atmosphere) light travels at slightly slower speeds and in media still denser, such as glass or plastic, it will travel slower still. The importance of this fact cannot be over emphasized. Without it, we could not see. With it, we can make and use lenses. It is with lenses that we image everything. It is through them that we are
Getting the Picture - the function of LensesTo see how a lens works there is just one more fundamental property of light that must be mentioned. Unlike the reasonable notion that light will slow down when it enters a medium denser than outer space, this other attribute is a little less intuitive.
When light travels between any two points it will always take the path that requires the least time. What this means is that light will not always follow the path representing the shortest distance. Most of the time the two paths are identical. But not always.
Let us turn on a light source, say a projector, and point it at the wall (or screen) across the room. The light rays making up the beam will reach the wall in the fastest time if they each travel in a straight line that we correctly define as the shortest distance between two points.
Now let's intersect the beam with a pane of glass so that all the light rays have to pass in and out of it before continuing their journey toward the wall. Once inside the new medium, the glass sheet, the light will have to slow down. But, remember, its task is nevertheless to get to that wall in the fastest possible time. So, in order to fulfill that assignment, each light ray will change its direction as it enters the glass such that it will move more closely to a line that is perpendicular to the surface of the pane. This line is called the normal and light will always be bent toward the normal when it enters a denser medium.
Conversely, when the light exits our pane of glass its direction will be bent away from the normal [Figure 2]. This is fortunate because, if the two surfaces of the glass sheet are parallel to one another, the consequence of the first bending is effectively canceled by the second. (This is why we can see through windows and windshields, etc. without perceptible distortion.) This process of bending or unbending the direction of light rays is called Refraction.
But what would happen if we arranged that the two surfaces of our pane of glass were not parallel to each other, if in fact the edge of the pane no longer looked like this ¡ and instead looked like £?
When that occurs, the normal to the first surface is not parallel to the normal to the second surface and very interesting things can be seen to happen when a beam of light is shined through it. Instead of being bent just once in one direction (and then being bent back), light rays passing though these two surfaces will be bent twice in the same direction. Rays subjected to this double bending will always exit the triangle in a direction away from the apex.
If we take the £ and add another one ¥ below the edge view of our glass will look like a diamond N. Light rays striking the bottom (and inverted) triangle will also be double bent and will, therefore, exit traveling away from their triangle's apex. This means that even if light rays arrive at the top and bottom of the first surface of the diamond-shaped glass in straight, parallel lines, they will exit the far sides of the diamond no longer parallel and will, at some point beyond it, intersect.
Once we recognize that simply by changing the angular relationship between two surfaces of a piece of glass or plastic we can alter and control the behavior of light rays passing through it, we have grasped the essential function of a lens.
Smoothing the edges of the diamond into a large variety of curves, making some convex, others concave, still others combinations of both [See Figure 3], will empower us to redirect light rays in a great many ways so that they may be made to converge (a positive lens), diverge (a negative lens). Note that all of the former are thicker in the middle than at their extremities and the latter are thinner at their centers.
The extent to which the direction of light rays can be modified by a single lens is dependent on three variables: the incident angle of the rays, the curvatures of the lens, and the material out of which the lens has been made. A material capable of transmitting light has what is called an index of refraction ("n") which is simply the quotient of the speed of light in a vacuum ("c") divided by whatever is the slower speed within the chosen material. Since "n" for a vacuum obviously equals 1, then the index of refraction for all other materials will be greater than 1. Since the index for air is only 1.00029, it is generally permissible to round it down to 1 in most calculations. Water has an index of 1.33 and glass is about 1.5. Interestingly enough, acrylic has a lower index than glass, about 1.4.
The significance of these indices is that the higher they become, the higher will be the optical density of the materials they describe. The precise relationship between any two materials with different indices was first established by the Dutch mathematician Willebrord Snell in 1621. Snell found that n1sinq1 = n2sinq2, where n1 and n2 are the two indices on either side of the surface and q1 is the angle of incidence to it and q2 the angle of refraction from it. Obviously, knowing any three of these variables permits the calculation of the fourth.
Now that we know how lenses work, let's take a closer look at how they image. We'll start by setting a vase full of cut flowers in front of a projection screen. After stipulating that there is plenty of illumination in the room, we will nevertheless notice that there is no "image" of a vase and flowers visible on the screen. The reason that this is so is that light rays reflected from all parts of our object (the vase of flowers) fall upon all parts of the screen equally. Thus, on the screen at least, there is no spatial distinction between one part of the object and another and, therefore, there cannot be any discernable image.
We see the vase of flowers because the aperture through which the light rays pass in our eyes is small enough that only a tiny subset of the rays emanating from our vase can be transmitted through it. Furthermore, of course, the curvature of the eye at that aperture causes light rays incident to it to be refracted such that they converge onto the screen of the retina behind.
To get a good image of the vase onto a full size projection screen, we want light rays from the vase (or a representation thereof) to be passed through a lens that will refract them so that they will subsequently all converge with no overlapping exactly at the plane of the screen. The more light rays we can get the lens to converge onto the screen (always with no overlapping), the brighter our image will be.
A line drawn through the two centers of curvature of a lens [see Figure 3] is called its principal axis. The point at which a beam of parallel light rays, traveling parallel to that axis will be made to converge is called the focal point. The distance from the center of a lens to its focal point is its focal length.
If we take a converging lens and place it before our vase so that the distance between them is less than the distance to the focal point, the image that we will see of the vase will be magnified and will appear right side up. This is called a virtual image.
When the distance between lens and object is greater than the focal point, the image will not be magnified and will be upside down. This is called a real image. Since our eyes continually look at objects well beyond their focal point, all the images reaching our retinas are inverted. Our brains, of course, cause this function of lenses to be continuously reverted. We close, therefore, by asking whether an image you see projected though lenses onto and from a screen is virtual or real, upside down or right side up?
In preparing this article, the author has principally utilized the following sources:
Asimov, Isaac, Understanding Physics: Light, Magnetism, and Electricity, New American Library, 1966.
Feynman, Richard P., Leighton, Robert B., and Sands, Matthew, The Feynman Lectures on Physics, Addison-Wesley Publishing Company, 1977.
Hewitt, Paul G., Conceptual Physics, Harper Collins, 7th edition, 1993.