1. Rules of refraction for convex lenses

Three generalisations ("rules") apply for the refraction of light rays through convex lenses whose surfaces are equally curved. Such lenses have two focal points F1 and F2, on each side of the lens, at the same distance from the optical centre (that is, F1 = F2).

  1. Any ray which travels parallel to the principal axis will be refracted and pass through the focal point, F2, of the lens. (Line X-Y in the above diagram).
  2. An incident ray passing through the focal point, F1, will be refracted and emerge in a direction parallel to the principal axis. (Line M-N)
  3. A ray which passes through the optical centre O of the lens will continue as if it had not been refracted. (Line A-B). (Click to trace the light rays)

2. Refraction by convex lenses

2.1 Case 1: The rays incident to the lens are parallel

All the incident rays will be refracted through the lens, and pass through the focal point F2. The light rays are said to be FOCUSSED at the focal point F2 (Click to trace the light rays).

This is what occurs when parallel rays from the sun are focussed on a plane surface by a magnifying glass. An intense bright spot (an image of the sun) appears at the plane surface, if the latter is at the focal point of the lens. (Wear sunglasses if you try this out!). Note, by the way, that heat rays (infrared rays) are also focussed at that point on the surface, which rapidly starts to smoulder, if it is made of combustible material.

2.2 Case 2: The object is more than twice the focal length away from the lens

We can determine the position and orientation of the image by drawing three rays coming from the top of the object, such that the rules of refraction through a converging lens are obeyed. We note that the image is INVERTED (upside down), and SMALLER than the object. (Click to trace the light rays). This image is a REAL IMAGE in that light rays are actually passing through it. Such an image can be projected onto a screen.

2.3 Case 3: The object lies at 2F1

A real, inverted image is formed at 2F2. No magnification takes place. (Click to trace the light rays).

2.4 Case 4: The object lies between 2F1 and F1

A real, inverted and magnified image is formed beyond 2F2. (Click to trace the light rays).

2.5 Case 5: The object lies at F1

If the object is relatively large, all rays emerge from the lens as a parallel bundle. No image is formed. (Click to trace the light rays).

If the object is small relative to the lens, all the rays will emerge form the lens so as to pass through F1. (Click to trace the light rays).

2.6 Case 6: The object lies between F1 and the lens

The rays DIVERGE from the lens on the side opposite to the object. Thus, no real image is formed. However, if one projects the rays towards the object side of the lens, they will converge to form a VIRTUAL IMAGE. This image is ERECT and MAGNIFIED. The image is virtual since no light rays actually pass through it. (Click to trace the light rays)

3. Refraction by concave lenses

All rays passing through a concave lens diverge upon leaving the lens. Only virtual images can be formed, on the same side of the lens as the object. A beam of parallel rays will appear to come from the focal point F1. (Click to trace the light rays)

4. Magnification

When an image is formed, the image may be larger, equal to, or smaller than the object. The MAGNIFICATION, m, that is obtained by the lens is given by:

The above holds only for so-called "thin lenses", where the thickness of the lens may be neglected in calculations.

4.1 The power of a lens

Optometrists frequently refer to the "power of a lens", a quantity that is measured in DIOPTERS, D. The power, P, of a lens is simply the reciprocal of the focal length of the lens. Thus, if the focal length of a lens is f, the power of the lens will be 1/f diopters. Diopters are non-SI units, and are simply reciprocal meters, m-1.

5. The lens formula

The focal length, f, the object distance, u, and the image distance, v, are related by the LENS FORMULA:

Distances to virtual images are negative, and if v = 0, no image is formed.

6. Additional questions