# Thin Lenses

When light passes from one medium to another, it is bent. We call this refraction. We make use of an object called a lens to bend light in a particular direction. Lenses take advantage of the fact that light is bent with relation to the normal of the surface of the lens not only as it enters the lens, but also again as it leaves. Lenses will cause light to converge or diverge, depending upon the construction of the lens. And while there are many different types constructed for various purposes, we will concentrate on converging lenses and diverging lenses.

There are (of course) some equations and terms that we need to be familiar with. A thin lens is one in which the thickness of the lens is small as compared to the overall radius of the lens. A line that passes through the center of the lens is called the principle axis. If several parallel lines enter a lens, they will all converge at a point called the focal point (F). The length from the center of the lens to the focal point is called the focal length (f). The focal length is the same on both sides of the lens. In lenses, we use a special notation called the Power (P) of a lens. The power is given by the inverse of the focal length, and is measured in Diopters (D).

The units of Diopters are meters-1. Thus, a lens with a focal length of 20 cm would have a power of 5 Diopters. Optometrists and ophthalmologists work with Diopters. A converging lens (sometimes called a positive lens) is a lens that is thicker in the middle than on the edges. The lens above is a converging lens. Converging lenses can produce real or virtual images. Ray tracing allows us to identify image distances and sizes. There are three useful rays that will allow us to determine where an image lies in a converging lens:

1. From object, parallel to principal axis. Passes through lens to rear focal point.
2. Ray passes from object straight through center of lens.
3. Ray from object passes through front focal point F1 and emerges parallel to the principle axis.

A real image is produced when the image is on the other side of the lens as the object. In this case, the image will be inverted. A virtual image is produced when the image is on the same side of the lens as the object. Here, the image will be upright.

Diverging lenses (also sometimes called negative lenses) will produce only a virtual image (the image forms on the same side of the lens as the object). The image will be upright, but the size of the image will be reduced. Converging lenses also have three rays which are useful to draw:

1. From object, parallel to principal axis. Appears to pass from focal point in front of the lens.
2. Ray passes from object straight through center of lens.
3. Ray from object passes through front focal point Fand appears to come from a line parallel to the principal axis.

To determine where an image forms and how big it is, we use the Lensmaker's Equation,

where do is the distance from the object to the center of the lens, di is the distance from the image to the center of the lens, and f is the focal length. Magnification (m) is given by the equation

where ho is the height of the object and hi is the height of the image. A negative magnification denotes the image is inverted with respect to the object. Determining where an image is from the equations is a matter of getting the sign convention correct.

• The focal length is positive for converging lenses and negative for diverging lenses.
• The object distance, do is positive if it is on the side of the lens from which the light is coming, otherwise, it is negative.
• The image distance, di is positive if it is on the opposite side of the lens from where the light is coming, otherwise, di is negative. di is positive for a real image, but negative for a virtual image.
• The height of the image, hi is positive if the image is upright with respect to the object and negative if it is inverted.

When two or more lenses are used together, the image is found, first by finding the image from the first lens, then using that image as the object for the second lens. If the image from the first lens appears behind the second lens, then it is treated as a virtual object. The overall magnification of the system is the product of the individual magnifications for each of the lenses. Compound microscopes and refracting telescopes are examples
of optical instruments that use combinations of lenses. Corrected vision (needing glasses) does the same thing - a second lens is placed in front of the lens of the eye in order to make the image focus on the retina.

• People who can only see distant objects in focus have a condition called hyperopia, or farsightedness. The image appears to form behind the retina. A converging lens is used to cause the image to form sooner.
• People who can only see close objects in focus have myopia, or nearsightedness. The image forms in front of the retina, and a diverging lens is used to form the image farther back in the eye.

For more on the eye, try http://www.physicsclassroom.com/Class/refrn/u14l6a.html