## Lens combinations

3.5.1 A general method

We can calculate the position and size of the final image produced by two lenses, in the following way. We start by finding the position and size of the image created by the first lens. This image becomes the object for the second lens. It can be a real or virtual object, depending on whether it is situated in front of or behind the second lens. The position and size of the final object are then calculated. The process may be repeated for any number of lenses.

3.5.2 Examples â lenses in contact

Two thin lenses in contact have focal lengths + f1 and + f2 respectively. What is the focal length (F) of the combination?

Note the emphasis that both focal lengths are positive, i.e. we have two converging lenses. A parallel beam strikes the first lens as illustrated in Figure 3.15. This will be focused at a distance f1 by the first lens and serve as a virtual point

(1) (2) final lens image 1st lens image

(1) (2) final lens image 1st lens image

object for the second lens. It is a virtual object since the light rays never reached that point, but were intercepted by the second lens.

By definition, final image distance = focal length of combination.

Applying the lens equation â + â =1 to lens (2):

u v f u = -fâ (virtual object distance) v = F (final image distance) f = f2 (focal length of lens under consideration)

If one of the lenses is divergent we insert a negative value for the focal length f for that lens.

### 3.5.3 The power of a lens

The shorter the focal length of a lens, the closer to itself it can focus a beam, i.e. the greater the change in the direction of light rays it produces.

It is logical therefore to define the power P of a lens as inversely proportional to the local length:

The unit of lens power is the diopter, defined as the power of a lens of focal length 1 m.

For thin lenses in contact the power of the combination is simply the sum of the powers of the individual components, remembering that the numerical value of P is negative if the lens is divergent.

Lenses and, to a lesser extent, mirrors are the central components of optical instruments. The most fundamental such instrument is the human eye, which brings the incoming light to a focus, performs a preliminary analysis of the signal, and then transmits the information on to the brain.

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