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very close together, and the light which is supposed to come in a straight line
1s spread out at an angle, so that when 1% comes into the eye 1% comes in from
sevoral directions. Also you wïll 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 ïn 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.
--- Trang 468 ---
20-6 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 in Eig. 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 A4 to Ö, say
by hitting the mirror. The iau for that chance is the following very strange one.
Take any path and fnd the time for that path; then make a complex number,
or draw a little complex vector, øe?, œbose œngle 9 is proportional to the time.
The number o 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 diferent 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 endl
Fig. 26-14. The summation of probability amplitudes for many neigh-
boring paths.
Now let us show how this implies the principle of least time for a mirror. WWe
consider all rays, all possible paths AD, AHB, AC, etc., in Eig. 26-3. The
path A4AJD2 makes a certain small contribution, but the next path, 1⁄5, takes
a quite diferent time, so its angle Ø is quite diferent. Let us say that point Œ
corresponds to minimum time, where 1Ý 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 geb near point Π(Fig. 26-14). So the arrows which we have to add are
coming almost exactly at the same angle for awhile near Œ, and then gradually
the time begins to increase again, and the phases go around the other way, and so
--- Trang 469 ---
on. Eventually, we have quite a tight knot. The total probability is the distanece
from one end to the other, squared. 4ửmost ølÏ oƒ that accwmnulated probabilit
occurs ïn the region tthere qÏl the œrrotus are ïn the sarmme đireciion (or in the same
phase). All the contributions from the paths which have very đjƒƑferent tỉìmes as
we change the path, cancel themselves out by pointing in diferent directions.
That is why, if we hide the extreme parts of the mirror, it still relects 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.
--- Trang 470 ---
Ấoormeofr'rcerÏl Ê)jpp££€-s
27-1 Introduction
In this chapter we shall discuss some elementary applications of the ideas of
the previous chapter to a number of practical devices, using the approximation
called geometrical optfics. Thïs is a most usefu]l approximation in the practical
design of many optical systems and instruments. Geometrical optics 1s either
very simple or else it is very complicated. By that we mean that we can either
study it only superficially, so that we can design instruments roughly, using rules
that are so simple that we hardly need deal with them here at all, since they
are practically of hipgh school level, or else, if we want to know about the small
errors in lenses and similar details, the subject gets so cormplicated that it is too
advanced to discuss herel TẾ one has an actual, detailed problem in lens design,
including analysis of aberrations, then he is advised to read about the subject
or else simply to trace the rays through the various surfaces (which is what the
book tells how to do), using the law of refraction from one side to the other,
and to ñnd out where they come out and see iŸ they form a satisfactory image.
People have said that this is too tedious, but today, with computing machines, it
is the right way to do it. One can set up the problem and make the calculation
for one ray after another very easily. 5o the subject is really ultimately quite
simple, and involves no new prineciples. Furthermore, it turns out that the rules of
either elementary or advanced optics are seldom characteristic of other fields, so
that there is no special reason to follow the subject very far, with one important
exception.
The most advanced and abstract theory of geometrical optics was worked
out by Hamilton, and it turns out that this has very important applications in
mechanics. Ïlt is actually even more important in mechanies than it is in opties,
and so we leave Hamilton”s theory for the subject ofadvanced analytical mechanies,
which is studied in the senior year or in graduate school. 5o, appreciating that
--- Trang 471 ---
Figure 27-1
geometrical optics contributes very little, except for its own sake, we now go on
to discuss the elementary properties of simple optical systems on the basis of the
principles outlined ïn the last chapter.
In order to go on, we must have one geometrical formula, which is the following:
1ƒ we have a triangle with a small altitude h and a long base đ, then the diagonal s
(we are goiïng to need it to fnd the difference in time between two different routes)
is longer than the base (Fig. 27-1). How mụuch longer? The diference A = s— đ
can be found in a number of ways. One way is this. W©e see that s2 — đ2 = h2,
or (s — đ)(s + đ) = h?. But s— đ= A, and s+d2s. Thus
A~ h2/2s. (27.1)
This is all the geometry we need to discuss the formation of images by curved
surfacesl
27-2 The focal length of a spherical surface
The first and simplest situation to discuss is a single refracting surface,
separating §wo media with diferent indices of refraction (Fig. 27-2). We leave
the case of arbitrary indices of refraction to the student, because deas are always
l6) vị €C ớ
AIR GLASS
Fig. 27-2. Focusing by a single refracting surface.
--- Trang 472 ---
the most Important thing, not the specifc situation, and the problem is easy
enough to do in any case. So we shall suppose that, on the left, the speed is 1