PHYS1330 Lenses
 Lenses (computer art by Michael Sargent)

Lenses and the Camera

The thin lens
A convex lens is a disk of glass or plastic fatter in the center than it is on the edge. It has the property of focusing parallel beams into a single point behind the lens through refraction.

Rays initially parallel to the axis converge on a focal point after passing through the lens. The convergence determines the necessary curvature of the lens cut. The distance behind the lens to the focal point is called the focal length of the lens. A fatter lens will have a shorter focal length.

In the thin-lens approximation, we do not need to consider the path through the lens material. We may assume the light bends only once in the plane of the lens. We are now prepared to see how a lens creates a real image of an object. Imagine light scattering off the top pine needle of a pine tree, and heading toward a lens.

Two critical rays allow one to determine where and what size the image will be:
1. A ray initially parallel to the axis will bend at the lens and pass through the focal point.
2. A ray initially passing through the forward focal point will bend at the lens and become parallel to the axis.
Remember that light paths are always reversible. If the light is parallel to the axis on one side, it must pass through the focal point on the other side. A third ray through the center of the lens does not bend (at least in the thin approximation) and confirms the convergence of the other two rays.

The image formed in this way is a real, rather than virtual, image. Light rays really do pass through the image and diverge out the other side just as they did from the original object. If your eye is in a position to capture the explanding rays, you will see a little upside-down tree in between you and the lens. Real images are usually inverted up/down and left/right, but not front/back. Real images can be recorded on film by placing the film where the image is located; virtual images cannot be recorded on film without turning them into a real image.

Image distance vs. object distance
If the object becomes closer to the lens, the image moves further away from the lens and becomes larger.

This diagram also shows that the real images are not reversed front/back; the two objects and their images are separated in the same direction in space. Here is an applet that allows one to adjust the distance and size of the object arrow by dragging on the tip of the the arrow. One can also change the focal length of the lens by dragging on the top or the bottom of the lens, but the action is less successful.

Depth of field
Where the image is focused behind the lens is called the focal plane of the image. When a camera changes focus from a distant object to a nearby object, the lens-to-film distance must increase. Technically, each object distance is focused at a different focal plane. However, if one moves a real film plane back and forth a bit, a range of object distances remains in acceptable focus. This range of acceptable distances is called the depth of field of a camera lens system. Only one object distance is in true focus, but other distances within the depth of field are not sufficiently out-of-focus at the chosen focal plane to be noticed.

One can change the depth of field by changing the aperture size of the lens. Consider the Example above with the entire lens admitting light. If one chooses to focus the camera on the far tree (i.e. place the film at the position of the far-tree image), the near tree will be very fuzzy since the light has not yet converged to its image at that plane. Now make the lens aperture much smaller (in the manner of making a pinhole camera aperture smaller).

The width of the non-converged beam for the near-tree image becomes much less, and can be made too small to notice. Then both trees will be in acceptable focus. Of course changing the aperture changes the brightness of the image, which will in turn change the exposure time needed to record an image on the film.

Control of the depth of field is very important for photographers who want to make one object (a flower, for example) stand out clearly against a background or foreground made fuzzy. The human eye also has a limited depth of field and uses the focusing information as a means of depth perception. So, a photograph with a sharply defined object among blurry objects appears to have some depth that is not really present on the flat print.

The magnifying glass
A magnifying glass is a simple convex lens. When one puts the object closer to the lens than the focal length, the lens makes a virtual, magnified image.

Two rays help locate the position and size of the image: the ray from the object running parallel to the axis, and the ray from the object running through the center of the lens. An observer looking at the rays emerging from the lens sees an upright, magnified image on the same side of the lens that the object is on. The observer does not see the object itself, unless part of it is visible beyond the edge of the lens.

Fresnel lenses
It is difficult to manufacture, transport, and support in a mounting, a very large lens. The largest single lenses in use today are on a few astronomical telescopes, and they are only about one meter in diameter. In the 18th century, lighthouses used mirrors and/or lenses to redirect light emerging in all directions from a lantern into parallel beams sweeping across the water. By keeping the beam parallel, it could be seen from much greater distances. In 1822, the French scientist Augustin Jean Fresnel invented a segmented lens that allowed the gathering of 83% of the lantern light, making lighthouses much more efficient and bright. Not only was the Fresnel lens more efficient, it was also much easier to make, transport, and install since it came in small pieces. Here is a lighthouse site with information about lighthouse Fresnel lenses.

Fresnel lenses can be made as one piece with carefully cut glass or plastic. The interior glass of a lens does nothing for the optical focusing. If one cuts out the interior and only maintains the surface curvature, one ends up with a much thinner, lightweight, focusing panel:

Traffic lights make use of such lenses. The course labKit contains a creditcard-sized Fresnel lens cast out of plastic.

Aberrations
As one might imagine, lenses are not perfect. One can cut a piece of fine quality glass into a shape that focuses monochromatic (single wavelength) light originally parallel to the axis into a single point behind the lens. However, to create an image of anything other than a distant star (the most parallel light available) one must also focus rays that are not parallel to the axis and/or have different wavelengths. No single piece of glass will do that. The various deformities of imaging are called aberrations. There are many kinds (see the title link), but we will consider only these:

Coma and spherical aberrations
A single lens designed to properly focus rays that are initially parallel to the axis into a point will show coma for angled rays. Coma looks sort of like a comet; any off-axis point on the object gets blurred out in the image with a tail extending away from the central axis.

Coma can be reduced by shaping the lens with spherical segment surfaces. More rays see similar curvature, and produce sharper images. However, points on or near the axis now get blurred in the image. The outer parts of a spherical-cut lens focus parallel rays closer to the lens than do the more central parts. The best single lenses juggle a compromise between coma and spherical aberrations.

Chromatic aberration
All materials suffer some dispersion, so the short-wave image usually focuses closer to the lens that the long-wave image. The human eye suffers from chromatic aberration. However, we are seldom presented with bright blue-violet objects that need focusing on, so the normal eye is designed to focus mid- and long wavelengths. As a result, at night it is difficult to focus on distant, blue neon lights. The short waves from these lights focus before they reach the retina and the eye cannot thin the lens enough to accomodate them.

Astigmatism
In eyes, different parts of the cornea-lens combination may focus the image in different places, or the curvature may be different in one direction vs. another. All such irregularities related to asymmetric curvature are called astigmatism. The writer of these pages (lsa-h) sees two images of the world separated by about 1/8 of a degree of angle (1/4 of the Moon's diameter). One image comes from the more central on-axis parts of my lens system, the other from the off-center parts. Both eyes are the same.

Field distortions
Parallel lines on the object may not be parallel on the image. Usually straight lines in the object translate to slightly curved lines in the image if they do not pass through the central axis. While images are highly distorted on the human retina, the brain has learned to compensate. Presumably fixed objects have a constant shape independent of where their images are focused on the retina. The brain has 'memorized' the conversion factors necessary to preserve this shape constancy. That memorization is something your brain does during the first few months of life when you can't talk.

Camera lenses
A single lens suffers from too many aberrations to do much better than a pinhole as a camera lens. Today's cameras use multiple lenses to correct these aberrations. For example, two lenses with different dispersion can be used to make two rays of different wavelengths converge to the same focal plane. Even more complications appear in a telephoto lens (a lens designed to produce the magnified image of a long focal-length lens in a short distance) or a zoom lens (a lens designed with an adjustable focal length). Here is a diagram of a particular zoom lens configuration.

Summary:
Real images are actual confluences of light rays that create in light a replication of the object. Virtual images are not actual confluences; rather, the lens or mirror re-directs rays to appear as if they originate at such a confluence.

The focal point of a lens is defined as that point at which rays from infinity (parallel rays) impinging on the lens are focused. The focal length is the distance of the focal point from the center of the lens. This distance is controlled by the curvature of the lens.

All cases of images from simple lenses/mirrors can be constructed geometrically from 2 or 3 important rays. One ray is the ray initially parallel to the axis, which ends up going through the focal point. Another ray is the ray which goes through the focal point and ends up going parallel to the axis. A third ray is the ray going through the center of the lens, which is not deflected. (For curved mirrors use similar rays with slightly different results defined by the law of reflection).

From simple lenses consisting of only one element, real images are generally inverted up-down and sideways, but not front-back.

Magnifying glasses are simple lenses used to produce a virtual image of an object by placing the object closer to the lens than the focal point.

All lenses suffer from a variety of aberrations. These "defects" may be corrected to some precision by employing multiple elements with different refraction and dispersion indices.

Sample questions for reflection

Ray tracing makes use of what property of light?

What is a real image? a virtual image?

What is the focal point of a lens? The focal length?

Be able to choose the correct ray from various alternatives passing through a lens.

Be able to choose the correct image location and size from various alternatives.

Is the image of a more distant object closer or further from the lens than that of a closer object?

What is meant by depth of field and how is it controlled in a camera?

What kind of image does a lens used as a magnifying glass produce?

Where does the object have to be to create the magnifying glass image?

How do fresnel lenses work?

Be able to identify various aberrations from their descriptions.

How do camera manufacturers correct for aberrations?