Converging and diverging lenses

A lens changes the direction of light passing through it. A converging lens brings parallel rays together, while a diverging lens spreads them apart.

The principal axis is the straight line that passes through the lens’s optical centre. The principal focus describes what happens to rays that are initially parallel to this axis:

• For a converging lens, the refracted rays meet at a real principal focus on the opposite side of the lens.
• For a diverging lens, the refracted rays spread out. If extended backwards, they meet at a virtual principal focus on the same side as the incoming light. The light itself does not pass through this focus.

A lens has a principal focus on each side, so the same description applies whichever direction light travels through it.

Focal length and lens power

The focal length is the distance from the optical centre of the lens to a principal focus. Under the usual sign convention, focal length $$f$$ is positive for a converging lens and negative for a diverging lens; its magnitude gives the distance.

Lens power measures how strongly a lens converges or diverges light:

Here $$f$$ is in metres and power $$P$$ is in dioptres, written $$\text{D}$$. A converging lens has positive power; a diverging lens has negative power. A lens with a larger power magnitude has a shorter focal length magnitude, so it bends parallel rays more strongly.

Tracing rays through lenses

A ray diagram uses the principal axis, optical centre and principal foci to show the paths of light rays. Two useful rules are:

• A ray through the optical centre is approximated as continuing without a change of direction.
• A ray initially parallel to the principal axis is refracted through the far principal focus by a converging lens. A diverging lens refracts it so that it appears to come from the principal focus on the incoming side.

For a diverging lens, the backwards extension to the virtual focus is a construction line, not the path of the light.

Thin-lens equation and magnification

A thin lens forms an image by refracting light from an object. The thin-lens equation relates the object’s distance from the lens, the image’s distance from the lens and the lens’s focal length:

Here, $$u$$ is the object distance, $$v$$ is the image distance and $$f$$ is the focal length. Measure the distances from the lens and use consistent units throughout the calculation.

Using the real-is-positive convention, a real object has a positive $$u$$, a real image has a positive $$v$$, and a virtual image has a negative $$v$$. A converging lens has a positive $$f$$. The sign of the calculated $$v$$ therefore tells us whether the image is real or virtual.

Linear magnification

Linear magnification compares image size with object size. When signed distances are used, the size ratio is:

For a real object and a real image, both distances are positive, so the size ratio can be written as $$m=v/u$$. A magnification greater than 1 means the image is larger than the object; a magnification less than 1 means it is smaller.

Thin-lens calculations

For a converging lens with $$f=10\text{ cm}$$ and an object at $$u=15\text{ cm}$$, substitution gives:

Thus $$v=30\text{ cm}$$. The positive distance identifies a real image. Its magnification is $$m=30/15=2$$, so it is twice the size of the object.

If the object is instead placed at $$u=5\text{ cm}$$ from the same lens:

Now $$v=-10\text{ cm}$$, so the image is virtual. Its size magnification is $$m=|-10/5|=2$$: it is twice the size of the object.

Finding the image with a ray diagram

A ray diagram shows the paths of light rays from an object as they pass through the lens. A real image forms at the position where the refracted rays meet. A virtual image is located where the backward extensions of the refracted rays meet: the rays appear to come from that position, but do not actually meet there.

Myopia and its correction

In myopia, or short-sightedness, nearby objects can be seen clearly but distant objects appear blurred. When the eye is relaxed, it brings light from a distant object to a focus in front of the retina, rather than on it. The light has spread out again by the time it reaches the retina, producing a blurred image.

This can happen because the eye is too long or because its optical power is too great, so it bends light too strongly. The far point is the greatest distance from the eye at which an object can be seen clearly without correction.

Correction with a diverging lens

A diverging lens placed in front of the eye spreads out incoming light before it enters the eye. The eye then bends the light so that it focuses on the retina.

For a very distant object, the incoming rays are approximately parallel. The correcting lens makes them diverge as though they came from a point at the eye’s far point. This point is a virtual image: the rays do not actually meet there, but their backward extensions do. The eye can focus these rays on the retina because they appear to come from a distance the eye can see clearly.

In a ray diagram, draw parallel rays entering the diverging lens. After the lens, draw the rays spreading apart and extend them backwards to the far point. Then show the eye focusing the rays on the retina.

Power of the correcting lens

Suppose a thin correcting lens is close to the eye and the far point is a distance $$R$$ metres away. For a very distant object, the object distance tends to infinity, $$u \to \infty$$. The lens must form a virtual image at the far point, so its image distance is $$v=-R$$. The negative sign indicates that the virtual image is on the incoming-light side of the lens.

For this distant object, the image is at the focal point of the lens. Its focal length $$f$$ and required power $$P$$ are therefore:

The focal length is measured in metres. The negative power identifies the correcting lens as a diverging lens.

Astigmatism and lens prescriptions

An eye can bend light by different amounts in different directions. This is called astigmatism. It commonly occurs because the cornea or the lens has different curvatures in different directions.

Each direction through the eye can be described as a meridian. When the eye has different optical powers in different meridians, it does not bring light to a single sharp focus in the same way in every direction. Vision can therefore appear blurred or distorted.

A cylindrical correction compensates by providing different lens powers in different meridians.

Reading an astigmatism prescription

A prescription commonly gives three values for each eye:

• Sphere (SPH): the spherical lens power, measured in dioptres ($$\mathrm{D}$$).
• Cylinder (CYL): the additional cylindrical power, also in dioptres. Its positive or negative sign is part of the prescribed power.
• Axis: the meridian, stated in degrees from $$0^\circ$$ to $$180^\circ$$, along which the cylindrical component adds no power.

The two principal meridians of the correction are at right angles. Along the axis, the power is SPH alone. In the meridian perpendicular to the axis, the cylindrical power is added:

For example, a prescription of SPH $$-2.00,\mathrm{D}$$, CYL $$-1.00,\mathrm{D}$$ and axis $$180^\circ$$ has a power of $$-2.00,\mathrm{D}$$ along $$180^\circ$$. Perpendicular to this, at $$90^\circ$$, the power is $$-2.00 + (-1.00) = -3.00,\mathrm{D}$$.