Lenses are made of transparent material such as glass or plastic with an index of refraction greater than that of air. At least one of the faces is a part of a sphere; a convex lens is thicker at the center than the edges, and a concave lens is thicker at the edges than the center. Convex lenses are called converging lenses, because they refract parallel light rays so that they meet. They are one of the most useful and important parts of all optical devices, and are found in eyeglasses, telescopes, microscopes, magnifying glasses, cameras and many other objects.
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Both the mirror equation and the magnification equation also apply to lenses. However, when dealing with lenses, the mirror equation is renamed the Lens Equation.
Double convex lenses have focal points on both sides of the lens, but it is also necessary to use points at twice the focal length to locate objects and images. Therefore, along the principal axis, there are points identified as F and as 2F on both sides of the lens.
As with mirrors, we only need to trace two rays in order to locate the image for lenses. Both rays change direction while inside the lens, and their convergence point on the opposite side of the lens is the image location. As can be seen in the figure above, Ray 1 approaches the lens parallel to the principal axis and is refracted through the focal point on the other side. Ray 2 travels through the focal point and is then refracted parallel to the principal axis. The yellow arrow on the right of the lens is the inverted image.
The diagram above shows the situation when the object is outside 2F. In this situation, the image will be between F and 2F on the other side and will be inverted, diminished, and real. A real image can be projected on a screen. That is, if you placed a sheet of paper at the image position, the image would actually appear on the paper.
If the object is placed between 2F and F, the image will appear beyond 2F on the other side. The image will be real, inverted, and enlarged. You can do a ray tracing like the one shown to demonstrate this is true.
If the object is placed inside F (between F and the lens), the image will be on the same side of the lens as the object and it will be virtual, upright, and enlarged.
In the sketch below, the object is red and has been placed inside F. The ray that approaches the mirror parallel to the principal axis is dotted yellow. It refracts through the focal point, also shown in dotted yellow. The ray that approaches the mirror through the focal point is dotted blue and refracts parallel to the principal axis, also shown in dotted blue. As you can see, the refracted rays diverge, so there will be no real image. If the eye is placed beyond the object around the 2F shown in the sketch, the eye will see the rays as if they have traveled in a straight line. These imaginary rays will converge at the tip of the green arrow which is the image position.
We have already learned that a lens is a carefully ground or molded piece of transparent material that refracts light rays in such a way as to form an image. Lenses serve to refract light at each boundary. As a ray of light enters a lens, it is refracted; and as the same ray of light exits the lens, it is refracted again. The net effect of the refraction of light at these two boundaries is that the light ray has changed directions. Because of the special geometric shape of a lens, the light rays are refracted such that they form images. Before we approach the topic of image formation, we will investigate the refractive ability of converging and diverging lenses.
First lets consider a double convex lens. Suppose that several rays of light approach the lens; and suppose that these rays of light are traveling parallel to the principal axis. Upon reaching the front face of the lens, each ray of light will refract towards the normal to the surface. At this boundary, the light ray is passing from air into a more dense medium (usually plastic or glass). Since the light ray is passing from a medium in which it travels fast (less optically dense) into a medium in which it travels relatively slow (more optically dense), it will bend towards the normal line. This is the FST principle of refraction. This is shown for two incident rays on the diagram below. Once the light ray refracts across the boundary and enters the lens, it travels in a straight line until it reaches the back face of the lens. At this boundary, each ray of light will refract away from the normal to the surface. Since the light ray is passing from a medium in which it travels slow (more optically dense) to a medium in which it travels fast (less optically dense), it will bend away from the normal line; this is the SFA principle of refraction.
The above diagram shows the behavior of two incident rays approaching parallel to the principal axis. Note that the two rays converge at a point; this point is known as the focal point of the lens. The first generalization that can be made for the refraction of light by a double convex lens is as follows:
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Now suppose that the rays of light are traveling through the focal point on the way to the lens. These rays of light will refract when they enter the lens and refract when they leave the lens. As the light rays enter into the more dense lens material, they refract towards the normal; and as they exit into the less dense air, they refract away from the normal. These specific rays will exit the lens traveling parallel to the principal axis.
The above diagram shows the behavior of two incident rays traveling through the focal point on the way to the lens. Note that the two rays refract parallel to the principal axis. A second generalization for the refraction of light by a double convex lens can be added to the first generalization.
Refraction Rules for a Converging LensThese two "rules" will greatly simplify the task of determining the image location for objects placed in front of converging lenses. This topic will be discussed in the next part of Lesson 5. For now, internalize the meaning of the rules and be prepared to use them. As the rules are applied in the construction of ray diagrams, do not forget the fact that Snells' Law of refraction of light holds for each of these rays. It just so happens that geometrically, when Snell's Law is applied for rays that strike the lens in the manner described above, they will refract in close approximation with these two rules. The tendency of incident light rays to follow these rules is increased for lenses that are thin. For such thin lenses, the path of the light through the lens itself contributes very little to the overall change in the direction of the light rays. We will use this so-called thin-lens approximation in this unit. Furthermore, to simplify the construction of ray diagrams, we will avoid refracting each light ray twice - upon entering and emerging from the lens. Instead, we will continue the incident ray to the vertical axis of the lens and refract the light at that point. For thin lenses, this simplification will produce the same result as if we were refracting the light twice.
Now let's investigate the refraction of light by double concave lens. Suppose that several rays of light approach the lens; and suppose that these rays of light are traveling parallel to the principal axis. Upon reaching the front face of the lens, each ray of light will refract towards the normal to the surface. At this boundary, the light ray is passing from air into a more dense medium (usually plastic or glass). Since the light ray is passing from a medium in which it travels relatively fast (less optically dense) into a medium in which it travels relatively slow (more optically dense), it will bend towards the normal line. This is the FST principle of refraction. This is shown for two incident rays on the diagram below. Once the light ray refracts across the boundary and enters the lens, it travels in a straight line until it reaches the back face of the lens. At this boundary, each ray of light will refract away from the normal to the surface. Since the light ray is passing from a medium in which it travels relatively slow (more optically dense) to a medium in which it travels fast (less optically dense), it will bend away from the normal line. This is the SFA principle of refraction. These principles of refraction are identical to what was observed for the double convex lens above.
The above diagram shows the behavior of two incident rays approaching parallel to the principal axis of the double concave lens. Just like the double convex lens above, light bends towards the normal when entering and away from the normal when exiting the lens. Yet, because of the different shape of the double concave lens, these incident rays are not converged to a point upon refraction through the lens. Rather, these incident rays diverge upon refracting through the lens. For this reason, a double concave lens can never produce a real image. Double concave lenses produce images that are virtual. This will be discussed in more detail in the next part of Lesson 5. If the refracted rays are extended backwards behind the lens, an important observation is made. The extension of the refracted rays will intersect at a point. This point is known as the focal point. Notice that a diverging lens such as this double concave lens does not really focus the incident light rays that are parallel to the principal axis; rather, it diverges these light rays. For this reason, a diverging lens is said to have a negative focal length.
The first generalization can now be made for the refraction of light by a double concave lens:
Now suppose that the rays of light are traveling towards the focal point on the way to the lens. Because of the negative focal length for double concave lenses, the light rays will head towards the focal point on the opposite side of the lens. These rays will actually reach the lens before they reach the focal point. These rays of light will refract when they enter the lens and refract when they leave the lens. As the light rays enter into the more dense lens material, they refract towards the normal; and as they exit into the less dense air, they refract away from the normal. These specific rays will exit the lens traveling parallel to the principal axis.
The above diagram shows the behavior of two incident rays traveling towards the focal point on the way to the lens. Note that the two rays refract parallel to the principal axis. A second generalization for the refraction of light by a double concave lens can be added to the first generalization.
Refraction Rules for a Diverging LensThe above discussion focuses on the manner in which converging and diverging lenses refract incident rays that are traveling parallel to the principal axis or are traveling through (or towards) the focal point. But these are not the only two possible incident rays. There are a multitude of incident rays that strike the lens and refract in a variety of ways. Yet, there are three specific rays that behave in a very predictable manner. The third ray that we will investigate is the ray that passes through the precise center of the lens - through the point where the principal axis and the vertical axis intersect. This ray will refract as it enters and refract as it exits the lens, but the net effect of this dual refraction is that the path of the light ray is not changed. For a thin lens, the refracted ray is traveling in the same direction as the incident ray and is approximately in line with it. The behavior of this third incident ray is depicted in the diagram below.
Now we have three incident rays whose refractive behavior is easily predicted. These three rays lead to our three rules of refraction for converging and diverging lenses. These three rules are summarized below.
Refraction Rules for a Converging Lens
These three rules of refraction for converging and diverging lenses will be applied through the remainder of this lesson. The rules merely describe the behavior of three specific incident rays. While there is a multitude of light rays being captured and refracted by a lens, only two rays are needed in order to determine the image location. So as we proceed with this lesson, pick your favorite two rules (usually, the ones that are easiest to remember) and apply them to the construction of ray diagrams and the determination of the image location and characteristics.
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