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In the first case, because the light
can hit anywhere on the drop, some of it is reflected in all directions from the
front surface. This
light forms a background to the rainbow, reducing the contrast and the
saturation of the colours.
The next diagram is a simulation of
the second case, in which the light is reflected once off the back of the drop –
Because the paper is only two-dimensional the simulation
is for a cylinder instead of a sphere, but the principle is the same.
The white line represents light going from right to left
and hitting one drop. It comes back in a cone of light. The
important point is that the edge of the cone is much brighter than the rest of
the volume. This is what makes the rainbow. The colours are seen
because the angle of the cone varies very slightly with the colour. The
effect of millions of drops together makes the rainbow that the picture
shows. The light coming back from a cloud represents the sum of the
results of the three cases shown in the diagram above.
Why does the cone have a bright
edge? The reason is that if we start with a ray of light that hits the
centre of the drop, and then look at rays hitting further and further from the
centre, the angle of the reflected light at first increases, but then it reaches a
maximum and then decreases.
This is shown in the next diagram –
The left hand side represents the
centre of a rain-drop, and the right hand side
the edge. The vertical axis represents the angle between the reflection
and the line to the sun.
Near the maximum, many different rays of light all give similar deflections –
that is why there is an arc of light centred on the extended line from the sun
to the observer. To
see this in action download the program or run from the current location,
by clicking on the diagram above. This is a basic principle in science
and maths – the probability is greatest at a maximum or a
minimum. In the same way an oscillating object spends longer near
the extremes than the middle. If you want to photograph racing
cars or motorbikes, a good place to stand is on the outside of a
bend. Not only do you have a longer time, the transverse velocity is
at a minimum, leading to sharper images.
This
diagram represents the intensity of the light at different
angles of reflection, on a scale running from zero degrees to 45 degrees, left to right. Zero
degrees represents looking straight ahead, the centre of the rainbow circle.
The long, almost constant
part represents the illumination inside the rainbow. The spike represents
the rainbow. The blue and grey areas represent rays coming before and
after the maximum angle. The intensity drops to zero outside the
rainbow. In a real sky there is, of course, some light outside the rainbow
because of light reflected from the front of drops, and light which has
penetrated a little way into the cloud and been scattered out.
This
calculation ignores the variation in the reflection of light with angle of
incidence. In reality, less light gets into the drop at the large
angles, and so the refracted intensity is over-estimated towards the
right, though no by a large factor in the important range.
In
the next diagram, twentyone sets of data have been added, representing twentyone
different colours, from the extreme red to the extreme violet end of the visible
spectrum. Different shades of grey have been used to avoid
simulating all the colours. Real sunlight is, of course, so continuous
that it looks look a smooth spectrum. The varying heights of the peaks
result from the finite resolution of the sums. We see that towards the
violet end of the spectrum, the colours are merged with many others, and in fact
we never see blue or violet. Even green is not easy to see in a rainbow.
This
calculation takes no account of the variation of the intensity of sunlight
with wavelength, and of the variation of the sensitivity of the eye with
wavelength. The next diagram uses a crude simulation of the
variation, there being little point in absolute accuracy for this purpose,
especially as the solar spectrum is biassed when the sun is low, as it has
to be for much of a rainbow to be seen. In that case the violet and
blue light is strongly affected by scattering (Rayleight
scattering). It’s hardly surprising that a rainbow is dominated
by red and orange.
We
now see that the blue and purple are much reduced. So is the red,
but it doesn’t have to cope with a background of other colours, only with
general reflected light.
The
diagram above is a simulation of cases one
and two, front reflection and single internal reflection, added
together. Compare it with the photographs of actual rainbows below.
Compare
the diagram with the photograph of an actual rainbow. The second and third copy of the photograph have had the contrast
increased using image handling software. The second rainbow is just about visible to the left of the
main one, near the top of the picture, while supernumerary bows are faintly
visible just to the right of the main bow, also near the top of the picture.
The last two pictures show clearly that the sky is darker outside the rainbow
than within it, especially near the bottom.
This
picture shows what you don’t want to see on your holiday, but it does
show quite well the brightness variations near a rainbow, and also a
glimpse of the secondary bow.
The diagrams are speckled because
individual rays of light were selected using random numbers, rather than by an
attempt to cover the space uniformly. This is a very simple example of a
Monte Carlo simulation.
The third case,
with two internal reflections in the drops, produces a reversed
rainbow outside the main rainbow, with a bright area outside itself, and a dark
area inside the rainbow. This
rainbow is much weaker than the first one and is not always visible.
Because different colours are
refracted slightly differently, there is a different cone for each colour.
Each observer in a group sees her or his own rainbow, even though all are
looking at the same drops of water. The centre of a rainbow is on a line
joining the observer’s head to the sun. Sometimes from an aeroplane a
passenger can see a complete circle in the cloud below, surrounding the shadow
of the aircraft.
Finally there are some
"supernumerary" bows, which require wave theory for their explanation.
The
diagram below shows the alignment of the rainbow with the sun. The
rainbow looks like a perfect circle, but it is really a circular cone of
light rays. There is no actual circle where the rainbow exists,
though the drops of water that cause it lie where the cone intersects
the surface of a cloud. Even that statement is an approximation,
because we can see some distance into a cloud. We cannot always
see the complete potential arc, because there is no cloud along the
requisite direction. If the sun is very close to the horizon, we
can sometimes see almost a semicircle.
The
next picture is from a very small part of a negative.
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