# How a Telescope Works

Russ Palum, Eastman Kodak Company

A telescope makes objects that are far away look closer. It does this by making the image formed by your eye larger. To understand how a telescope does this some background is required. We will address,

1. What is the small angle approximation?
2. What does an "object at infinity" mean?
3. How does a lens image an object at infinity?
4. What determines the height of an object in the image plane of a lens?

Some technical liberties are taken,

1. Only geometric optics is considered, diffraction and aberrations are ignored.
2. Some definitions are used loosely, for example the principal ray definition is not strictly accurate.

## Small angle approximation

The small angle approximation is commonly used in optics. For small angles, the angle in radians is approximately the tangent of the angle. The angle in radians is the angle in degrees divided by 57.29. This is a good approximation up to 10 degrees and more. At 10 degrees, the tangent is 0.1762, and the angle in radians is 0.1745 radians.

## Object at infinity Figure1. Lens, image and object

The image of the object is formed here. As the illustration progresses the object point gets further away and the point the rays converge toward in the image plane gets closer to the lens. Notice the object ray angles get shallower (smaller) as the object gets further from the lens until the bundle of object rays are parallel to each other when the object is infinitely far away in the last illustration. The additional rays in red show that all rays that are parallel to each other converge to the same image point. In addition, as the object plane goes to infinity the image plane approaches (is asymptotic to) the focal length of the lens (f).

## Image formation and image height

A camera is shown taking a picture of a tree in figure 2. The rays diverge from their respective object points in all directions but the lens aperture is small and far away so, the rays reaching the lens from each object point are effectively a parallel bundle. Remember, all of the rays that are parallel to each other converge to one image point. Figure 2.

When the rays from one object point are all parallel to each other, as shown in figure 3,the image is formed one focal length from the lens. The bundle of rays from top of the tree and the bundle from the trunk are not parallel to each other. The angular difference between the two bundles of rays determines the height of the image point. The ray in the center of each bundle is the principal ray. The principal ray for an object on axis is the optical axis. A perfect lens with no distortion will image a point at a distance h from the optical axis where h= f*tan( ), f is the focal length of the lens and is the angle of the principal ray to the optical axis. Figure 3. Simple lens

## The eye

The eye, with some approximation, uses a simple lens to form an image on the retina (the retina is similar to the film or electronic imager used in a camera). The eye can focus on objects that are between about 10 inches and infinity. Figure 4 is a schematic of the eye showing a bundle of rays on axis and a bundle off axis. Notice the image on the retina is upside down. The upside down image is sorted out by experience; you don't know the image is upside down because it has always been upside down. To make an object seem closer the image formed by the eye must be larger. To make the image larger the angle between ray bundles from different object points has to be increased. Figure 4. The eye

## Infinite image and Infinite object

In the first case shown in figure 5 a lens forms an image of an infinite object point one focal length from the lens at a height h=f* tan( ). For small angles in radians, h= f * In the second case a lens images an object point at infinity because the object point is one focal length in front of the lens. The equation h=f*tan( ) must still be satisfied. Figure 5. Infinite object and an object one focal length from the lens.

## The telescope

With the background out of the way it only takes a few paragraphs to show how a telescope works. A telescope is a type of afocal converter, this means it is used to view objects at infinity and the image formed by the telescope appears to be at infinity. When you look through a telescope your eye adjusts its focus as if the objects are at a large distance. The telescope makes objects appear closer by increasing the angle of the ray bundles entering your eye. This is called angular magnification. A simple refractor telescope, shown in figures 6 and 7, consists of a long focal length lens followed by a short focal length lens. The long focal length lens forms an image of a distant object one focal length (f1) behind the lens and the short focal length lens, the eye lens, is placed one focal length (f2) behind the image formed by the first lens. The short focal length lens forms an image of the long focal length lens image at infinity. In the axial case shown in figure 6, a parallel bundle of rays enters the telescope from an object point on the optical axis at infinity and a parallel bundle of rays leaves the telescope, this bundle appears to be from a distant object point on the optical axis. Figure 6. Telescope with axial object Figure 7. Telescope off axis

An off axis point is shown in figure 7. In this case, for small angles,

 h1 = f1 * 1 h2 = f2 * 2 , h1 = h2 f1 * 1 f2 * 2 2/ 1 = f1 / f2

The telescope magnifies the angle by a factor of f1 / f2, this is called the magnification of a telescope (the X number marked on the side). This type of telescope is an astronomical telescope because it forms an image that is upside down and reversed right to left.

The clue that the image is reversed is in figure 7, notice the incoming bundle of rays rise going left to right and the exiting bundle falls going left to right. The reversed image is not a problem when looking at stars but the reversed image is a problem for gun sights and other types of terrestrial telescopes. Terrestrial telescopes use prisms or additional lenses to make the image right reading. Binoculars are a pair of telescopes that use prisms to make the images right reading.

Another disclaimer is required, the telescope, as shown in figure 7, requires a field lens at the intermediate image. The field lens does not change the size of the image, it bends the cone of rays produced by the objective lens so the light passes through the eye lens. Without the field lens the telescope will have a very small angle of view.

The telescope in figure 6 is called a refractor because refraction is used to form the image. Refraction is the bending of rays as light moves from one medium to another (for example, air to glass or glass to air). The Newtonian telescope is another type of astronomical telescope. Newton invented this telescope because early refractor telescopes suffered from chromatic aberration; different colors of light do not focus in the same plane. Newton solved this problem by using a curved mirror in place of the first lens in a refractor. Newton was convinced refractors would always suffer from chromatic aberration, but in 1757, John Dollond invented the achromatic lens. This lens is usually made by cementing two lenses together that are made from different types of glass. One lens corrects the aberrations of the other. Sometimes the two lenses have an air space between them.

Two other telescopes are common. The catadioptric telescope uses two mirrors and produces a right reading image. This design can be used as a terrestrial telescope or an astronomical telescope. Another telescope, the Galilean telescope is limited to about 4X magnification. This telescope uses a positive lens followed by a negative lens, the image in this case is right reading. Opera glasses usually use this design. A Galilean telescope can also be put in front of a camera lens to produce a larger image on the film effectively increasing the focal length of the lens. These are commonly called teleconverters.