waves rays Consider rays of light from an object being reflected by a plane mirror (the rays are diverging): mirror object

Optics

Light and its interaction with lenses and mirrors.

We assume that we can ignore the wave properties of light.

waves rays

We represent the light as rays, and ignore diffraction. We can do this as long as the ray, and the objects it reflects from/is refracted by, are much wider that the wavelength of the light.

Light from an object

Light from a single point on an object spreads out in all directions – it is diverging light.

Reflection from a flat mirror

Consider rays of light from an object being reflected by a plane mirror (the rays are diverging):

mirror

object image

Each ray obeys the usual law of reflection (θ

= θ

), and the rays continue to diverge at the same

rate. What happens when we look into the mirror? To the viewer, the rays of light appear to travel in

straight lines, and seem to come from an object behind the mirror. The light is identical to that which would have come from an object behind the mirror if the mirror was not present – we see an image of the object. Since the light rays do not actually come from the image, we call it a virtual image.

Diverging light appears to come from an object

Converging light

If rays of light are converging, they will all meet at a single point:

screen

If the rays are projected onto a screen, you can see an image of the original object on the screen – a real image.

diverging light from real object or virtual image

converging light forms a real image on a screen

How to change the divergence/convergence of light

A problem with trying to project real images is that light from objects always diverges. We need to have a way of changing the rate at which light converges or diverges. This can be done using a curved mirror (reflection) or a curved lens (refraction).

Since the surface is curved, rays of light hitting the mirror or lens in different places are reflected or refracted in different directions.

Diverging mirror

Converging mirror

Diverging lens

Converging lens

Lenses and mirrors

A curved surface changes the convergence/divergence of light. If we want to use these lenses or mirrors to form clear images of objects, they need to have the correct curvature – the surface should be parabolic ie shaped like a parabola. Unfortunately, it’s hard to make a good parabolic surface, so almost all lenses and mirrors are made with spherical surfaces. A sperical surface is almost exactly the same as a parabolic surface, as long as the light hits the lens or mirror near the centre.

EXTRA Since a spherical surface isn’t ideal, it produces distortions in the image – spherical aberration.

EXTRA Since the refractive index of glass varies with wavelength, different colours are refracted differently, producing chromatic aberration. A combination of different materials can be used in the lens to reduce this.

convex mirror diverging concave mirror converging lens thicker in middle converging

• bi-convex

• plano-convex

• meniscus (concave-convex)

lens thinner in middle diverging

• bi-concave

• plano-concave

• meniscus (concave-convex)

Focal length

A good way to measure the effect of a lens or mirror is to the focal length. This is the distance from the lens at which parallel rays of light will be focussed to a point. This point is called the focal point.

The rays coming from a diverging mirror or lens do not meet at a point, but they do appear to come from a point. We represent this by a negative focal length.

• converging lenses and mirrors have positive focal lengths

• diverging lenses and mirrors have negative focal lengths

• the points on each side of the lens where parallel rays will be focussed are the focal points

Ray diagrams

We can see what will happen in an optical system by drawing a diagram showing where the rays from an object will go – a ray diagram.

First, we draw a line through the centre of the optical system – the optic axis.

Secondly, we can draw some rays on the diagram. There are three easy rays to draw:

• a ray travelling through the centre of the lens continues in a straight line

• a ray parallel to the optic axis is deflected through the focal point

• a ray passing through the (other) focal point will be deflected so that it is parallel to the optic axis (rays in an optical system are reversible)

Using these principal rays, the position, size, and type of the image can be found. We only need to use

two of them; usually the first two are used. The image is at the point where the principal rays meet.

Example: real image produced by a converging lens

Example: virtual image produced by a diverging lens

There are three main cases of interest if we have a single lens:

1. real image produced by a converging lens (pictured above) – when the object is more than a focal length away from the lens

2. virtual image produced by a converging lens (pictured above) – when the object is less than one focal length away from the lens

3. virtual image produced by a diverging lens – object any distance away

Optical instruments

Some optical instruments consist of just a single lens - for example, a magnifying glass, a camera, or a lens in a pair of glasses. Sometimes, two lenses are needed.

Telescope

A telescope consists of a long focal length objective lens, and a shorter focal length eyepiece. The eyepiece is used to view and magnify the real image produced by the objective.

Microscope

A microscope uses a very short focal length objective lens, and a longer focal length eyepiece.

The eye

The eye can also considered an optical instrument. The focal length of the eye is variable. An real image is formed on the back of the eye, the retina, which is covered in light-sensitive receptor cells (rods and cones).

EXTRA Eye safety and lasers

The light from a bright object like the sun is not focussed to a single point - an image of the object is formed. The energy in the light from the object is spread out over a large area (compared to the rods and cones on the retina). The light from a laser is parallel, and will be focussed to a very small area. The size of the spot will only be limited by diffraction, and will be about a wavelength wide.

So a laser of of only moderate power will produce very high intensities on the retina and will cause

damage where the greater energy in, for example, sunlight, spread out over a much larger area, will

not.

Mathematical optics ADVANCED

While ray diagrams can be very useful for illustrating what happens in an optical system, they are not very accurate. For accuracy, we need to use mathematical methods.

Vergence

We measure how fast light is converging or diverging by it’s vergence:

V = ± ⁿ d

where n is the refractive index the light is travelling in (not the refractive index of the lens), and d is the distance to the point at which the rays of light meet – the point from which they are diverging from, or converging to. The sign (positive or negative) of the vergence is determined by whether the light is converging or diverging:

• Negative vergence is diverging light

Light from an object, diverging light produced by another lens (virtual image), or light that has been focussed to a real image and is diverging again

• Positive vergence is converging light

Light that has been made to converge by a lens or mirror – this light will form a real image

• Zero vergence is parallel light

Very distant objects will have a vergence of almost zero Vergence has SI units of m

1. This unit is often called a dioptre.

Power of a lens or mirror

The power of a lens or a curved measures the effect that it has on the convergence/divergence of light.

The power is related to the focal length:

P = ± ⁿ f The sign of the power depends on the type of lens:

• Negative power ⇒ diverging lense/mirror

Convex mirror, concave lens, lens thinner in the middle

• Positive power ⇒ converging lens/mirror

Concave mirror, convex lens, lens thicker in the middle

• Zero power

Flat mirror or flat sheet of glass

The SI units of power are also dioptres

Effect of a lens/mirror

V

= _P + _V

Note that:

• Initial vergence V

is usually negative, since light from objects always diverges

• A diverging lens/mirror (with a negative power P) will make the divergence even more nega- tive – the image will always be virtual

• A converging lens/mirror (with a positive power) can produce either a real image (V

> _{0) or} virtual image (V

< ₀₎

You may have done similar calculations using the formula 1

d

+ ¹

d

= ¹ f

or a similar formula. You can use such a formula if you want; note that the formula is the same, except for possibly different conventions for positive and negative signs.

Magnification

How large is the image compared to the object? The magnification depends on the position of the object and image:

m = ^V

V

• Positive magnification ⇒ upright image

• Negative magnification ⇒ inverted image

How to calculate the power

It’s useful to be able to calculate the power of a lens or a mirror if we don’t know the focal length.

Mirror

P = ± ²ⁿ r

where r is the radius of curvature of the mirror, and n is the refractive index of the surrounding

medium (usually n = 1). The sign depends on whether the mirror is converging (convex – positive)

or diverging (concave – negative).

Lens

P = ⁿ

− _n

r

where r is the radius of curvature, n

is the refractive index inside the curve (not necessarily inside the lens), and n

is the refractive index outside the curve.

This formula is called the lensmaker’s formula. It gives the power of a single curved surface. Usually a lens has two curved surfaces. You will need to use this formula twice, and add the two powers together.

More than one lens or mirror

If they are very close together, just add the powers. Otherwise, we just need to find the final ver-

gence V

produced by the first lens, then find the initial vergence V

at the second lens (the vergence

changes as the rays travel), and then find the effect of the second lens.