GEOMETRICAL OPTICS

1. In the previous lecture we learned that light is waves,
and that waves spread out from every point. But in
many circumstances we can ignore the spreading
(diffraction and interference), and light can then be
assumed to travel along straight lines as rays. This
is hown by the existence of sharp shadows, as for
case of the eclipse illustrated here.
2. When light falls on a flat surface, the angle of
incidence equals the angle of reflection. You
can verify this by using a torch and a mirror,
or just by sticking pins on a piece of paper in
front of a mirror. But what if the surface is not
perfectly flat? In that case, as shown in fig. (b),
the angle of incidence and reflection are equal
at every point, but the normal direction differs
from point to point. This is called "diffuse
reflection". Polishing a surface reduces the
diffusiveness.
3. If you look at an object in the mirror, you
will see its image. It is not the real thing, and
that is why it is called a "virtual" image. You
can see how the virtual image of a candle is
formed in this diagram. At each point on the
surface, there is an incident and reflected ray.
If we extend each reflected ray backwards,
it appears as if they are all coming from the
same point. This point is the image of the tip
of the flame. If we take other points on the
candle, we will get their images in just the
same way. This way we will have the image
of the whole candle. The candle and its
image are at equal distance from the mirror.
4. Here is another example of image formation. A source of light is placed in front of a
bi-convex lens which bends the light as shown. The eye receives rays of light which seem
to originate from a positon that is further away than the actual source.
Now just to make the point even more forcefully, in all three situations below, the virtual
image is in the same position although the actual object is in 3 different places.
5. Imagine that you have a sphere of radius R and that you can cut
out any piece you want. The outside or inside surface can be
silvered, as you want. You can make spherical mirrors in this
way. These can be of two kinds. In the first case, the silvering
can be on the inside surface of the sphere, in which case this is
is called a convex spherical mirror. The normal directed from
the shiny surface to the centre of the sphere (from which it
was cut out from) is called the principal axis, and the radius
of curvature is . The other situation is that in which the
ou
R
tside surface is shiny. Again, the principal axis the same,
but now the radius of curvature (by definition) is - . What
does a negative curvature mean? It means precisely what
has been illus
R
trated - a convex surface has a positive and a
concave surface has a negative curvature.
6. Now we come to the important point: a beam of parallel
rays will reflect off a convex mirror and all get together
(or converge) at one single point, called the focus. Now
look at the figure. Since the incident and reflected angles
are equal for every ray, you can see that the ray will have
to pass through the focus at F, at distance / 2. We say
that the focal length of such
R
a mirror is f =R/ 2.
7. The concave mirror is different. A parallel beam incident
from the left is reflected and the rays now spread out
(diverge). However, if we were to look at any outgoing
ray and extend it backward, all the rays would appear to
be coming from one single point. This is the
because there is, in fact, no real source of light there. From
the diagram you can see the vir
virtual focus
tual focus F is at distance
/ 2 the mirror. We say that the focal length of such
a mirror is / 2.
R behind
f= −R
8. Now let's see how images are formed by concave mirrors.
Take two rays that are emitted from the top of an object
placed in front of the mirror. Call them the P and F ray,
and see what happens to them after reflection. Where the
two cross, that is the image point. As you can see the
image is inverted and smaller than the actual object if the
object is far away (i.e. lies to the left of C) and is larger if
close to the mirror. The magnification of size makes this
useful as a shaving mirror, among other things.
9. Now repeat for a concave mirror. Here the rays will
never actually meet so we can have only a virtual
object. Take 3 rays - call them P,F, and C - and see
how they are reflected from the mirror's surface.
Where they meet is the position of the image. The
image is always smaller than the actual size of the
object. This is obviously useful for driving a car
because you can see a wide area.
f
Focusing light with lens
Convergent lens
f
f
( )
1 2
1 1 1 1 l n
f R R
⎛ ⎞
= − ⎜ − ⎟
⎝ ⎠
R2 < 0
R1 > 0
10. A lens is a piece of glass curved in a definite way.
Because the refractive index of glass is bigger than
one, every ray bends towards the normal. Here you
see a double convex lens that focuses a beam of
parallel rays. For a perfect lens, all the rays will
converge to one single point that is (again) called
the focus. The distance f is called the focal length.
Of course, light can equally travel the other way,
so if a point source is placed at the focus of a
convex lens then a parallel beam of light will
emerge from the other side. This is how some
film projectors produce a parallel beam.
A concave lens does not cause a parallel beam to
converge. On the contrary, it makes it diverge, as
shown. Note, however, that if the rays are
continued backwards, then they appear to come
from one single point, which is here the virtual
focus. The distance f is the focal length.
12. Here is how a concave (or divergent) lens forms an image. An observer on the side
opposite to the object will see the image upright and smaller in size than the object.
Divergent lens
Object
Virtual and upright image
1 2
2
13. A lens can be imagined as cut out of two spheres
of glass as shown, with the spheres having radii
R and R . Note that they tend to bend light in
opposite ways. By convention, R is
( )
1 2
negative. It
is possible to show that the focal length of the
lens is , 1 n 1 1 1 .
f R R
⎛ ⎞
= − ⎜ − ⎟
⎝ ⎠
14. The following figure summarizes the shape of some common types of lenses.
Note that a flat surface has infinite radius of curvature. The focal legth of each can be
calculated using the previous formula.
15. The "strength" of a lens is measured in diopters. If the focal length of a lens is expressed
in metres, the diopter of the lens is defined as 1/ . If the refractive index of the
glass in a lens is , then the diopters due to the first interf
D f
n
=
1
2 1 1 2 2 1 2
ace and the second interface
are, ( 1) / and (1 ) / . The total diopter of the lens is .
16. For any optical system - meaning a collection of lenses and mirrors - we can def
D
D D=n− R D= −n R D=D+D
ine a
magnification factor as a ratio of sizes -- see the diagram below.
17. The perfect lens will focus a parallel beam of rays all to exactly the same focus. But no
lens is perfect, and every lens suffers from aberration although this can be made quite
small by following one lens with another. Below you see an example of "spherical
aberration". Rays crossing different parts of the lens do not reach exactly the same focus.
This distorts the image. Computers can design lens surfaces to minimize this aberration.
Planar convex
R1 > 0
R2 = infinity
Bi-concave
R1 < 0
R2 > 0
Planar-concave
R1 = infinity
R2 > 0
CONVEX LENS CONCAVE LENS
Bi-convex
R1 > 0
R2 < 0
Object
Optical Element
Image
h h′
Lateral Magnification M = h′/h
Spherical Aberration
Principal