Mirrors

Every surface reflects some of the light that reaches it. Glass and the cornea send back a few per cent; a mirror, coated with a thin layer of metal, sends back almost all of it. Reflection is even simpler than refraction: there is no refractive index to worry about and no dependence on colour, just a single rule. Yet a curved mirror focuses light exactly as a lens does: it has a power, and it obeys the same vergence equation.

#The law of reflection

Here is a beam of light meeting a flat mirror. Drag the start of the ray to change its direction:

The light leaves on the other side of the normal at the same angle as it arrived: the angle of reflection ${\theta }_r$ equals the angle of incidence ${\theta }_i$. The wavefronts show why. The part of a wavefront that reaches the mirror first turns back first, and because light travels just as fast on the way out as on the way in, the returning wavefront makes the same angle with the mirror as the arriving one. Nothing in this depends on the material or on the colour of the light.

#Images in a plane mirror

Light from an object spreads out in every direction. After a flat mirror reflects it, it is still spreading out, but now as if it came from a point behind the mirror. Drag the object, or the eye:

Traced back, the reflected rays all meet at one point, the image. It lies as far behind the mirror as the object is in front of it. It is a virtual image: no light actually passes through it, which is why it can sit inside a wall. Wherever the eye is, it finds the image in the same place, as long as the mirror is big enough to send it some light.

A plane mirror has no power: light leaves it with the same vergence it arrived with. The image of a whole object is upright and the same size, but reversed from front to back, the parts of the object nearest the mirror being nearest in the image too. We see that as left and right swapped, which is why your reflection raises its left hand when you raise your right.

#Curved mirrors

Now curve the mirror, so that it is hollow toward the light: a concave mirror. Here is a single, flat wavefront approaching one. Move it with the slider:

The edges of the wavefront meet the mirror before its centre does, and they turn back first. While the centre travels the last stretch $s$ to the middle of the mirror, the edges are already travelling the same distance back. When the centre arrives, the edges are ahead of it by $2s$, twice the depth of the mirror, and the returning wavefront is curved. It closes in on one point, the focal point F.

How strongly is it curved? A shallow curve of radius $r$ is, at a height $y$ from the axis, about $s=y^2/2r$ deep. A shallow mirror sends the light back almost along the axis, so the reflected wavefront is twice as deep, $2s=y^2/r$, which is the depth of a curve of radius $r/2$. So reflected light converges to a point $r/2$ in front of the mirror, halfway to its centre of curvature C, and the mirror has a power of

with $r$, the size of the radius, in metres. A concave mirror with a radius of 40 cm has a power of +5.00 D and focuses parallel light 20 cm in front of it. A converging glass surface of the same radius has only $(1.5-1)/0.4=+1.25\text{D}$: the mirror is four times as strong, because instead of bending light a little, it turns it right round.

#Concave and convex

A mirror that bulges toward the light, a convex mirror, does the opposite. The centre of the wavefront reaches it first, and the reflected light spreads out as if from a focal point behind the mirror. Its power is negative, $P=-2/r$. Try both:

Concave mirrors converge light, like plus lenses; convex mirrors make it diverge, like minus lenses. A mirror has just one focal point, because the reflected light travels back toward the side it came from. And since reflection doesn’t depend on refractive index, a mirror focuses every colour in the same place: it has no chromatic aberration. That is one reason the largest telescopes use mirrors rather than lenses.

#The vergence equation for mirrors

A mirror changes the vergence of light just as a lens does, so the vergence equation of Chapter 4 works unchanged:

The only thing to remember is that the light turns round. Distances count as positive in the direction the light travels: toward the mirror before it reflects, away from it afterwards. Light from an object 60 cm in front of a +5.00 D mirror arrives with $U=-1.67\text{D}$ and leaves with $V=+3.33\text{D}$, converging to a real image 30 cm in front of the mirror, on the same side as the object. Drag the object:

The easy rays of Chapter 4 work here too. A ray parallel to the axis reflects through F, a ray through F reflects parallel to the axis, and a ray through the centre of curvature meets the mirror square on and comes straight back. A fourth, aimed at the middle of the mirror, leaves at the same angle on the other side of the axis. The magnification is $U/V$, as for a lens.

Bring the object inside F and the image becomes virtual, upright and magnified, behind the mirror: this is a shaving or make-up mirror. A convex mirror always gives a small, upright, virtual image of a wide field of view, which is why it is used for car wing mirrors and for watching the corners of shops.

#Spheres and parabolas

The curved mirrors so far have been parabolic, which is why they focused perfectly however wide they were. A spherical mirror, the easiest kind to make, is not quite perfect. Widen this one:

Rays near the axis meet at F, but rays from the rim of a spherical mirror cross the axis a little closer to the mirror than F. This is spherical aberration, just as for a lens in Chapter 5. The perfect shape for focusing parallel light is a parabola, which sends every ray through F. Near its centre a parabola is almost exactly a sphere, so $P=2/r$ holds for both. The difference matters for mirrors that are wide compared with their radius, such as satellite dishes and searchlight reflectors, and for those that must be accurate to a fraction of a wavelength, such as telescope mirrors.

#The cornea as a mirror

The front surface of the cornea reflects about 2.5% of the light that reaches it: that is the bright point of light you see in someone’s eye. To that light, the cornea is a convex mirror. With a radius of about 7.8 mm, its power is $-2/0.0078\approx -256\text{D}$, so a lamp in front of the eye forms a small, upright, virtual image just behind the cornea, called the first Purkinje image. Its size depends on the radius of the cornea, and that is how a keratometer measures it:

A keratometer shines two lights, its mires, at the eye from about 75 mm away, and a telescope measures how far apart their images are. Because the cornea’s power is so much greater than the vergence of the arriving light, the images lie close to its focal point, and the magnification is about

where $h$ is the separation of the two mires, $h^{\prime }$ that of their images and $d$ the distance to the eye. The radius follows from the size of the image: steeper corneas give smaller images. Notice also that the light used is reflected from two points about 3 mm apart, not from the very centre of the cornea. The eye never keeps perfectly still, so real keratometers use a prism to double the image and measure it by bringing the two copies edge to edge, which does not depend on where the image is.

The result is given as a power, using the keratometric index 1.3375 instead of the cornea’s true index of 1.376:

A radius of 7.80 mm reads 43.27 D. The lower index makes rough allowance for the weak minus power of the back of the cornea, so that the reading approximates the power of the whole cornea rather than the +48 D of its front surface alone (Chapter 2), and it makes a radius of 7.50 mm read exactly 45.00 D. Measured in two meridians, the readings give the corneal astigmatism of Chapter 7: 7.80 mm horizontally and 7.65 mm vertically read 43.27 D and 44.12 D, about 0.85 D of with-the-rule astigmatism.

#Recap

• The angle of reflection equals the angle of incidence. A plane mirror forms a virtual image as far behind it as the object is in front.
• A curved mirror has a power of $P=\pm 2/r$, positive for concave and negative for convex mirrors, with its focal point halfway to its centre of curvature.
• The vergence equation $V=U+P$ applies, with distances measured in the direction the light travels. Real images form in front of the mirror.
• A wide spherical mirror has spherical aberration; a parabolic mirror brings parallel light to a perfect focus.
• The cornea is a convex mirror of about −256 D. The size of the image it forms gives its radius, and $K=337.5/r$ with $r$ in millimetres.

#Test yourself

#Further reading

• M. H. Freeman and C. C. Hull, Optics. Butterworth-Heinemann.
• M. P. Keating, Geometric, Physical, and Visual Optics. Butterworth-Heinemann.
• R. B. Rabbetts, Bennett and Rabbetts’ Clinical Visual Optics. Butterworth-Heinemann.