26 Optics: The Principle of Least Time
26–1 Light
This is the first of a number of chapters on the subject of
electromagnetic radiation. Light, with which we
see, is only one small part of a vast spectrum of the same kind of
thing, the various parts of this spectrum being distinguished by
different values of a certain quantity which varies. This variable
quantity could be called the “wavelength.” As it
varies in the visible spectrum, the light apparently changes color from
red to violet. If we explore the spectrum systematically, from long
wavelengths toward shorter ones, we would begin with what are usually
called radiowaves. Radiowaves are technically available in a wide
range of wavelengths, some even longer than those used in regular
broadcasts; regular broadcasts have wavelengths corresponding to about
500 meters. Then there are the so-called “short waves,” i.e., radar
waves, millimeter waves, and so on. There are no actual boundaries
between one range of wavelengths and another, because nature did not
present us with sharp edges. The number associated with a given name for
the waves are only approximate and, of course, so are the names we give
to the different ranges.
Then, a long way down through the millimeter waves, we come to what we
call the infrared, and thence to the visible
spectrum. Then going in the other direction, we get into a region which
is called the ultraviolet. Where the ultraviolet
stops, the x-rays begin, but we
cannot define precisely where this is; it is roughly at 10−8 m, or
10−2 μm. These are “soft” x-rays; then there are ordinary
x-rays and very hard x-rays; then γ-rays, and so on, for smaller
and smaller values of this dimension called the wavelength.
Within this vast range of wavelengths, there are three or more regions
of approximation which are especially interesting. In one of these, a
condition exists in which the wavelengths involved are very small
compared with the dimensions of the equipment available for their
study; furthermore, the photon energies, using the quantum theory, are
small compared with the energy sensitivity of the equipment. Under
these conditions we can make a rough first approximation by a method
called geometrical optics. If, on the other hand, the
wavelengths are comparable to the dimensions of the equipment, which is
difficult to arrange with visible light but easier with radiowaves, and
if the photon energies are still negligibly small, then a very useful
approximation can be made by studying the behavior of the waves, still
disregarding the quantum mechanics. This method is based on the
classical theory of electromagnetic radiation, which will be
discussed in a later chapter. Next, if we go to very short wavelengths,
where we can disregard the wave character but the photons have a very
large energy compared with the sensitivity of our equipment,
things get simple again. This is the simple photon picture, which
we will describe only very roughly. The complete picture, which unifies
the whole thing into one model, will not be available to us for a long
time.
In this chapter our discussion is limited to the geometrical optics
region, in which we forget about the wavelength and the photon
character of the light, which will all be explained in due time. We do
not even bother to say what the light is, but just find out
how it behaves on a large scale compared with the dimensions of
interest. All this must be said in order to emphasize the fact that
what we are going to talk about is only a very crude approximation;
this is one of the chapters that we shall have to “unlearn”
again. But we shall very quickly unlearn it, because we shall almost
immediately go on to a more accurate method.
Although geometrical optics is just an approximation, it is of very
great importance technically and of great interest historically. We
shall present this subject more historically than some of the others
in order to give some idea of the development of a physical theory or
physical idea.
First, light is, of course, familiar to everybody, and has been
familiar since time immemorial. Now one problem is, by what process do
we see light? There have been many theories, but it finally
settled down to one, which is that there is something which enters the
eye—which bounces off objects into the eye. We have heard that idea
so long that we accept it, and it is almost impossible for us to
realize that very intelligent men have proposed contrary
theories—that something comes out of the eye and feels for the
object, for example. Some other important observations are that, as
light goes from one place to another, it goes in straight
lines, if there is nothing in the way, and that the rays do not seem
to interfere with one another. That is, light is crisscrossing in all
directions in the room, but the light that is passing across our line
of vision does not affect the light that comes to us from some
object. This was once a most powerful argument against the corpuscular
theory; it was used by Huygens.
If light were like a lot of arrows shooting along, how could other
arrows go through them so easily? Such philosophical arguments are not
of much weight. One could always say that light is made up of arrows
which go through each other!
26–2 Reflection and refraction
Fig. 26–1. The angle of incidence is equal to the angle of reflection.
The discussion above gives enough of the basic idea of
geometrical optics—now we have to go a little further into the
quantitative features. Thus far we have light going only in straight
lines between two points; now let us study the behavior of light when
it hits various materials. The simplest object is a mirror, and the
law for a mirror is that when the light hits the mirror, it does not
continue in a straight line, but bounces off the mirror into a new
straight line, which changes when we change the inclination of the
mirror. The question for the ancients was, what is the relation
between the two angles involved? This is a very simple relation,
discovered long ago. The light striking a mirror travels in such a way
that the two angles, between each beam and the mirror, are equal. For
some reason it is customary to measure the angles from the normal to
the mirror surface. Thus the so-called law of reflection is
(26.1)
That is a simple enough proposition, but a more difficult problem is
encountered when light goes from one medium into another, for example
from air into water; here also, we see that it does not go in a
straight line. In the water the ray is at an inclination to its path
in the air; if we change the angle θi so that it comes down
more nearly vertically, then the angle of “breakage” is not as
great. But if we tilt the beam of light at quite an angle, then the
deviation angle is very large. The question is, what is the relation
of one angle to the other? This also puzzled the ancients for a long
time, and here they never found the answer! It is, however, one of the
few places in all of Greek physics that one may find any experimental
results listed. Claudius
Ptolemy
made a list of the angle in water for each of a number of
different angles in air. Table 26–1 shows the angles in the
air, in degrees, and the corresponding angle as measured in the water.
(Ordinarily it is said that Greek scientists never did any experiments.
But it would be impossible to obtain this table of values without
knowing the right law, except by experiment. It should be noted,
however, that these do not represent independent careful measurements
for each angle but only some numbers interpolated from a few
measurements, for they all fit perfectly on a parabola.)
Fig. 26–2. A light ray is refracted when it passes from one medium into
another.
Table 26–1.
This, then, is one of the important steps in the development of
physical law: first we observe an effect, then we measure it and list
it in a table; then we try to find the rule by which one thing
can be connected with another. The above numerical table was made in
140 A.D., but it was not until 1621 that someone finally
found the rule connecting the two angles! The rule, found by
Willebrord Snell, a Dutch
mathematician, is as follows: if θi is the angle in air and
θr is the angle in the water, then it turns out that the sine
of θi is equal to some constant multiple of the sine of θr:
(26.2)
Table 26–2.
26–3 Fermat’s principle of least time
Now in the further development of science, we want more than just a
formula. First we have an observation, then we have numbers that we
measure, then we have a law which summarizes all the numbers. But the
real glory of science is that we can find a way of
thinking such that the law is evident.
The first way of thinking that made the law about the behavior of
light evident was discovered by
Fermat
in about 1650, and it is called the principle of least
time, or Fermat’s principle. His idea
is this: that out of all possible paths that it might take to get from
one point to another, light takes the path which requires the
shortest time.
Let us first show that this is true for the case of the mirror, that
this simple principle contains both the law
of straight-line propagation and the law for the mirror. So, we are
growing in our understanding! Let us try to find the solution to the
following problem. In Fig. 26–3 are shown two points,
A and B, and a plane mirror, M M′. What is the way to get from
A to B in the shortest time? The answer is to go straight from
A to B! But if we add the extra rule that the light has
to strike the mirror and come back in the shortest time, the
answer is not so easy. One way would be to go as quickly as possible
to the mirror and then go to B, on the path A D B. Of course, we
then have a long path D B. If we move over a little to the right,
to E, we slightly increase the first distance, but we greatly
decrease the second one, and so the total path length, and
therefore the travel time, is less. How can we find the point C for
which the time is the shortest? We can find it very nicely by a
geometrical trick.
Fig. 26–3. Illustration of the principle of least time.
We construct on the other side of M M′ an artificial point B′,
which is the same distance below the plane M M′ as the point B is
above the plane. Then we draw the line E B′. Now because B F M is a
right angle and B F=F B′, E B is equal to E B′. Therefore the sum
of the two distances, A E+E B, which is proportional to the time it
will take if the light travels with constant velocity, is also the sum
of the two lengths A E+E B′. Therefore the problem becomes, when is
the sum of these two lengths the least? The answer is easy: when the
line goes through point C as a straight line from A
to B′! In other words, we have to find the point where we go toward the
artificial point, and that will be the correct one. Now if A C B′ is a
straight line, then angle B C F is equal to angle B′ C F and thence to
angle A C M. Thus the statement that the angle of
incidence equals
the angle of reflection is equivalent to the
statement that the light goes to the mirror in such a way that it comes
back to the point B in the least possible time. Originally, the
statement was made by Hero of Alexandria that the light travels in such
a way that it goes to the mirror and to the other point in the shortest
possible distance, so it is not a modern theory. It was this that
inspired Fermat to suggest to
himself that perhaps refraction operated on a similar basis. But for
refraction, light obviously does not use the path of shortest
distance, so Fermat tried
the idea that it takes the shortest time.
Before we go on to analyze refraction, we should make one more remark
about the mirror. If we have a source of light at the point B and it
sends light toward the mirror, then we see that the light which goes
to A from the point B comes to A in exactly the same manner as
it would have come to A if there were an object at B′, and
no mirror. Now of course the eye detects only the light which
enters it physically, so if we have an object at B and a mirror
which makes the light come into the eye in exactly the same manner as
it would have come into the eye if the object were at B′, then the
eye-brain system interprets that, assuming it does not know too much,
as being an object at B′. So the illusion that there is an
object behind the mirror is merely due to the fact that the light
which is entering the eye is entering in exactly the same manner,
physically, as it would have entered had there been an object
back there (except for the dirt on the mirror, and our knowledge of
the existence of the mirror, and so on, which is corrected in the
brain).
Now let us demonstrate that the principle of
least time will give Snell’s law of
refraction. We must, however, make an assumption about the speed of
light in water. We shall assume that the speed of light in water is
lower than the speed of light in air by a certain factor, n.
Fig. 26–4. Illustration of Fermat’s principle for refraction.
Fig. 26–5. The minimum time corresponds to point C, but nearby points
correspond to nearly the same time.
In Fig. 26–4, our problem is again to go from A to B
in the shortest time. To illustrate that the best thing to do
is not just to go in a straight line, let us imagine that a beautiful
girl has fallen out of a boat, and she is screaming for help in the
water at point B. The line marked x is the shoreline. We are at
point A on land, and we see the accident, and we can run and can
also swim. But we can run faster than we can swim. What do we do? Do
we go in a straight line? (Yes, no doubt!) However, by using a little
more intelligence we would realize that it would be advantageous to
travel a little greater distance on land in order to decrease the
distance in the water, because we go so much slower in the
water. (Following this line of reasoning out, we would say the right
thing to do is to compute very carefully what should be done!)
At any rate, let us try to show that the final solution to the problem
is the path A C B, and that this path takes the shortest time of all
possible ones. If it is the shortest path, that means that if we take
any other, it will be longer. So, if we were to plot the time it takes
against the position of point X, we would get a curve something like
that shown in Fig. 26–5, where point C corresponds to
the shortest of all possible times. This means that if we move the
point X to points near C, in the first approximation there
is essentially no change in time because the slope is zero at
the bottom of the curve. So our way of finding the law will be to
consider that we move the place by a very small amount, and to demand
that there be essentially no change in time. (Of course there is an
infinitesimal change of a second order; we ought to have a
positive increase for displacements in either direction from C.) So
we consider a nearby point X and we calculate how long it would take
to go from A to B by the two paths, and compare the new path with
the old path. It is very easy to do. We want the difference, of
course, to be nearly zero if the distance X C is short. First, look
at the path on land. If we draw a perpendicular X E, we see that this
path is shortened by the amount E C. Let us say we gain by not having
to go that extra distance. On the other hand, in the water, by drawing
a corresponding perpendicular, C F, we find that we have to go the
extra distance X F, and that is what we lose. Or, in time, we
gain the time it would have taken to go the distance E C, but we lose
the time it would have taken to go the distance X F. Those times must
be equal since, in the first approximation, there is to be no change
in time. But supposing that in the water the speed is 1/n times as
fast as in air, then we must have
(26.3)
Therefore we see that when we have the right point, X C sinE X C=n⋅X C sinX C F or, cancelling the common hypotenuse length X C
and noting that
we have
(26.4)
So we see that to get from one point to another in the least time when
the ratio of speeds is n, the light should enter at such an angle
that the ratio of the sines of the angles θi and θr is
the ratio of the speeds in the two media.
26–4 Applications of Fermat’s principle
Now let us consider some of the interesting consequences of the
principle of least time. First is the
principle of reciprocity. If to go from A
to B we have found the path of the least time, then to go in the
opposite direction (assuming that light goes at the same speed in any
direction), the shortest time will be the same path, and therefore, if
light can be sent one way, it can be sent the other way.
Fig. 26–6. A beam of light is offset as it passes through a transparent
block.
An example of interest is a glass block with plane parallel faces, set
at an angle to a light beam. Light, in going through the block from a
point A to a point B (Fig. 26–6) does not go through
in a straight line, but instead it decreases the time in the block by
making the angle in the block less inclined, although it loses a
little bit in the air. The beam is simply displaced parallel to itself
because the angles in and out are the same.
Fig. 26–7. Near the horizon, the apparent sun is higher than the true
sun by about 1/2 degree.
A third interesting phenomenon is the fact that when we see the sun
setting, it is already below the horizon! It does not look as
though it is below the horizon, but it is (Fig. 26–7). The
earth’s atmosphere is thin at the top and dense at the bottom. Light
travels more slowly in air than it does in a vacuum, and so the light of
the sun can get to point S beyond the horizon more quickly if, instead
of just going in a straight line, it avoids the dense regions where it
goes slowly by getting through them at a steeper tilt. When it appears
to go below the horizon, it is actually already well below the horizon.
Another example of this phenomenon is the mirage that one often sees
while driving on hot roads. One sees “water” on the road, but when he
gets there, it is as dry as the desert! The phenomenon is the following.
What we are really seeing is the sky light “reflected” on the road:
light from the sky, heading for the road, can end up in the eye, as
shown in Fig. 26–8. Why? The air is very hot just above
the road but it is cooler up higher. Hotter air is more expanded than
cooler air and is thinner, and this decreases the speed of light less.
That is to say, light goes faster in the hot region than in the cool
region. Therefore, instead of the light deciding to come in the
straightforward way, it also has a least-time path by which it goes into
the region where it goes faster for awhile, in order to save time. So,
it can go in a curve.
Fig. 26–8. A mirage.
As another important example of the principle of least time, suppose that we would like to arrange a
situation where we have all the light that comes out of one point,
P, collected back together at another point, P′
(Fig. 26–9). That means, of course, that the light can go in
a straight line from P to P′. That is all right. But how can we
arrange that not only does it go straight, but also so that the light
starting out from P toward Q also ends up at P′? We want to bring
all the light back to what we call a focus. How? If the light
always takes the path of least time, then certainly it should not want
to go over all these other paths. The only way that the light can be
perfectly satisfied to take several adjacent paths is to make those
times exactly equal! Otherwise, it would select the one of least
time. Therefore the problem of making a focusing system is merely to
arrange a device so that it takes the same time for the light to go on
all the different paths!
Fig. 26–9. An optical “black box.”
This is easy to do. Suppose that we had a piece of glass in which
light goes slower than it does in the air (Fig. 26–10). Now
consider a ray which goes in air in the path P Q P′. That is a longer
path than from P directly to P′ and no doubt takes a longer time.
But if we were to insert a piece of glass of just the right thickness
(we shall later figure out how thick) it might exactly compensate the
excess time that it would take the light to go at an angle! In those
circumstances we can arrange that the time the light takes to go
straight through is the same as the time it takes to go in the
path P Q P′. Likewise, if we take a ray P R R′ P′ which is partly inclined, it
is not quite as long as P Q P′, and we do not have to compensate as much
as for the straight one, but we do have to compensate somewhat. We end
up with a piece of glass that looks like Fig. 26–10. With
this shape, all the light which comes from P will go to P′. This, of
course, is well known to us, and we call such a device a converging
lens. In the next chapter we shall actually calculate what shape
the lens has to have to make a perfect focus.
Fig. 26–10. A focusing optical system.
Take another example: suppose we wish to arrange some mirrors so that
the light from P always goes to P′ (Fig. 26–11). On
any path, it goes to some mirror and comes back, and all times must be
equal. Here the light always travels in air, so the time and the
distance are proportional. Therefore the statement that all the times
are the same is the same as the statement that the total distance is
the same. Thus the sum of the two distances r1 and r2 must be a
constant. An ellipse is that curve which has the property that
the sum of the distances from two points is a constant for every point
on the ellipse; thus we can be sure that the light from one focus will
come to the other.
Fig. 26–11. An ellipsoidal mirror.
The same principle works for gathering the light of a star. The great
200-inch Palomar telescope is built on the following
principle. Imagine a star billions of miles away; we would like to
cause all the light that comes in to come to a focus. Of course we
cannot draw the rays that go all the way up to the star, but we still
want to check whether the times are equal. Of course we know that when
the various rays have arrived at some plane K K′, perpendicular to
the rays, all the times in this plane are equal
(Fig. 26–12). The rays must then come down to the mirror and
proceed toward P′ in equal times. That is, we must find a curve which
has the property that the sum of the distances X X′+X′ P′ is a
constant, no matter where X is chosen. An easy way to find it is to
extend the length of the line X X′ down to a plane L L′. Now if we
arrange our curve so that A′ A′′=A′ P′, B′ B′′=B′ P′, C′ C′′=C′ P′, and so on, we will have our curve, because then of course, A A′+A′ P′=A A′+A′ A′′ will be constant. Thus our curve is the locus of all
points equidistant from a line and a point. Such a curve is called a
parabola; the mirror is made in the shape of a parabola.
Fig. 26–12. A paraboloidal mirror.
The above examples illustrate the principle upon which such optical
devices can be designed. The exact curves can be calculated using the
principle that, to focus perfectly, the travel times must be exactly
equal for all light rays, as well as being less than for any other
nearby path.
We shall discuss these focusing optical devices further in the next
chapter; let us now discuss the further development of the
theory. When a new theoretical principle is developed, such as the
principle of least time, our first
inclination might be to say, “Well, that is very pretty; it is
delightful; but the question is, does it help at all in understanding
the physics?” Someone may say, “Yes, look at how many things we can
now understand!” Another says, “Very well, but I can understand
mirrors, too. I need a curve such that every tangent plane makes equal
angles with the two rays. I can figure out a lens, too, because every
ray that comes to it is bent through an angle given by Snell’s
law.” Evidently the statement of least time
and the statement that angles are equal on reflection, and that the
sines of the angles are proportional on refraction, are the same. So is
it merely a philosophical question, or one of beauty? There can be
arguments on both sides.
However, the importance of a powerful principle is that it
predicts new things.
It is easy to show that there are a number of new things predicted by
Fermat’s principle. First, suppose that
there are three media, glass, water, and air, and we perform a
refraction experiment and measure the index n for one medium against
another. Let us call n12 the index of air (1) against water
(2); n13 the index of air (1) against glass (3). If we
measured water against glass, we should find another index, which we
shall call n23. But there is no a priori reason why there
should be any connection between n12, n13, and n23. On
the other hand, according to the idea of least time, there is a
definite relationship. The index n12 is the ratio of two things,
the speed in air to the speed in water; n13 is the ratio of the
speed in air to the speed in glass; n23 is the ratio of the speed
in water to the speed in glass. Therefore we cancel out the air, and
get
(26.5)
In other words, we predict that the index for a new pair of
materials can be obtained from the indexes of the individual
materials, both against air or against vacuum. So if we measure the
speed of light in all materials, and from this get a single number for
each material, namely its index relative to vacuum, called ni
(n1 is the speed in vacuum relative to the speed in air, etc.),
then our formula is easy. The index for any two materials i and j
is
(26.6)
Another argument for the principle of least
time, another prediction, is that if we measure the speed of
light in water, it will be lower than in air. This is a prediction of
a completely different type. It is a brilliant prediction, because all
we have so far measured are angles; here we have a theoretical
prediction which is quite different from the observations from which
Fermat deduced the idea of least time. It
turns out, in fact, that the speed in water is slower than the
speed in air, by just the proportion that is needed to get the right
index!
26–5 A more precise statement of Fermat’s principle
Actually, we must make the statement of the principle of least time a little more accurately. It was not stated
correctly above. It is incorrectly called the
principle of least time and we have gone
along with the incorrect description for convenience, but we must now
see what the correct statement is. Suppose we had a mirror as in
Fig. 26–3. What makes the light think it has to go to
the mirror? The path of least time is clearly A B. So some
people might say, “Sometimes it is a maximum time.” It is not
a maximum time, because certainly a curved path would take a still
longer time! The correct statement is the following: a ray going in a
certain particular path has the property that if we make a small
change (say a one percent shift) in the ray in any manner whatever,
say in the location at which it comes to the mirror, or the shape of
the curve, or anything, there will be no first-order change in
the time; there will be only a second-order change in the
time. In other words, the principle is that
light takes a path such that there are many other paths nearby which
take almost exactly the same time.
The following is another difficulty with the principle of least time, and one which people who do not like this
kind of a theory could never stomach. With Snell’s theory we can
“understand” light. Light goes along, it sees a surface, it bends
because it does something at the surface. The idea of causality, that
it goes from one point to another, and another, and so on, is easy to
understand. But the principle of least time
is a completely different philosophical principle about the way nature
works. Instead of saying it is a causal thing, that when we do one
thing, something else happens, and so on, it says this: we set up the
situation, and light decides which is the shortest time, or the
extreme one, and chooses that path. But what does it do,
how does it find out? Does it smell the nearby paths,
and check them against each other? The answer is, yes, it does, in a
way. That is the feature which is, of course, not known in geometrical
optics, and which is involved in the idea of wavelength; the
wavelength tells us approximately how far away the light must
“smell” the path in order to check it. It is hard to demonstrate
this fact on a large scale with light, because the wavelengths are so
terribly short. But with radiowaves, say 3-cm waves, the distances
over which the radiowaves are checking are larger. If we have a source
of radiowaves, a detector, and a slit, as in Fig. 26–13,
the rays of course go from S to D because it is a straight line,
and if we close down the slit it is all right—they still go. But now
if we move the detector aside to D′, the waves will not go through
the wide slit from S to D′, because they check several paths
nearby, and say, “No, my friend, those all correspond to different
times.” On the other hand, if we prevent the radiation from
checking the paths by closing the slit down to a very narrow crack,
then there is but one path available, and the radiation takes it! With
a narrow slit, more radiation reaches D′ than reaches it with a wide
slit!
Fig. 26–13. The passage of radiowaves through a narrow slit.
One can do the same thing with light, but it is hard to demonstrate on
a large scale. The effect can be seen under the following simple
conditions. Find a small, bright light, say an unfrosted bulb in a
street light far away or the reflection of the sun in a curved
automobile bumper. Then put two fingers in front of one eye, so as to
look through the crack, and squeeze the light to zero very gently. You
will see that the image of the light, which was a little dot before,
becomes quite elongated, and even stretches into a long line. The
reason is that the fingers are very close together, and the light
which is supposed to come in a straight line is spread out at an
angle, so that when it comes into the eye it comes in from several
directions. Also you will notice, if you are very careful, side
maxima, a lot of fringes along the edges too. Furthermore, the whole
thing is colored. All of this will be explained in due time, but for
the present it is a demonstration that light does not always go in
straight lines, and it is one that is very easily performed.
26–6 How it works
Finally, we give a very crude view of what actually happens, how the
whole thing really works, from what we now believe is the correct,
quantum-dynamically accurate viewpoint, but of course only
qualitatively described. In following the light from A to B in
Fig. 26–3, we find that the light does not seem to be in
the form of waves at all. Instead the rays seem to be made up of
photons, and they actually produce clicks in a photon counter, if we
are using one. The brightness of the light is proportional to the
average number of photons that come in per second, and what we
calculate is the chance that a photon gets from A to B, say
by hitting the mirror. The law for that chance is the following
very strange one. Take any path and find the time for that path; then
make a complex number, or draw a little complex vector, ρ ei θ, whose angle θ is proportional to the
time. The number of turns per second is the frequency of the
light. Now take another path; it has, for instance, a different time,
so the vector for it is turned through a different angle—the angle
being always proportional to the time. Take all the available
paths and add on a little vector for each one; then the answer is that
the chance of arrival of the photon is proportional to the square of
the length of the final vector, from the beginning to the end!
Fig. 26–14. The summation of probability amplitudes for many neighboring
paths.
Now let us show how this implies the principle of least time for a mirror. We consider all rays, all
possible paths A D B, A E B, A C B, etc., in Fig. 26–3. The
path A D B makes a certain small contribution, but the next path, A E B,
takes a quite different time, so its angle θ is quite different.
Let us say that point C corresponds to minimum time, where if we
change the paths the times do not change. So for awhile the times do
change, and then they begin to change less and less as we get near
point C (Fig. 26–14). So the arrows which we have to add are
coming almost exactly at the same angle for awhile near C, and then
gradually the time begins to increase again, and the phases go around
the other way, and so on. Eventually, we have quite a tight knot. The
total probability is the distance from one end to the other, squared.
Almost all of that accumulated probability occurs in the region
where all the arrows are in the same direction (or in the same phase).
All the contributions from the paths which have very different
times as we change the path, cancel themselves out by pointing in
different directions. That is why, if we hide the extreme parts of the
mirror, it still reflects almost exactly the same, because all we did
was to take out a piece of the diagram inside the spiral ends, and that
makes only a very small change in the light. So this is the relationship
between the ultimate picture of photons with a probability of arrival
depending on an accumulation of arrows, and the principle of least time.
- Although it can be deduced if the additional assumption is made that adding a layer of one substance to the surface of another does not change the eventual angle of refraction in the latter material. ^