Diverging lenses

What a diverging lens does

• A diverging (concave) lens is thinner in the middle than at its edges.
• Rays arriving parallel to the principal axis are refracted so that they spread apart.
• Traced backward, those spreading rays appear to come from a single point on the same side as the object — the principal focus of a diverging lens.
• Because no light actually passes through that point, it is called a virtual focus, and the focal length is taken as negative:

$$f < 0 \text{ for a diverging lens}$$

The construction rays

The rules mirror those for a converging lens, with the focus on the near side:

• Ray 1 — parallel in, appears to come from $F$. A ray parallel to the principal axis is refracted so that it travels as if it had come from the focus on the near side. Draw the outgoing ray solid, and its backward extension to $F$ dashed.
• Ray 2 — through the optical centre. A ray aimed at the centre of the lens passes straight through, undeviated.
• The two refracted rays diverge, so extend them backward as dashed lines until they meet. That meeting point is the top of the image.

The image is always the same

Unlike a converging lens, a diverging lens gives the same kind of image for every object position:

• virtual — the rays never actually cross,

• upright,

• smaller than the object (diminished),

• located between the lens and the focus, on the same side as the object.

• There is no object position that produces a real or inverted image from a single diverging lens.

• This is exactly the behaviour of a convex mirror, for the same reason: both are diverging optics.

Calculations

• The same equations apply, with $f$ negative:

$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \qquad m = \frac{h_i}{h_o} = \frac{d_i}{d_o}$$

• With $f$ negative and $d_o$ positive, $\dfrac{1}{d_i} = \dfrac{1}{f} - \dfrac{1}{d_o}$ is always negative, so $d_i$ is always negative — the algebra guarantees the virtual image.
• The magnification always comes out less than 1.

Where they are used

• Spectacles for short sight (myopia). A short-sighted eye focuses light too early, in front of the retina. A diverging lens spreads the light slightly before it enters the eye, moving the focus back onto the retina.
• Peepholes in doors, and as part of camera and telescope lens systems, where they correct the focusing of the converging lenses they are paired with.