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close to the axis will still come to the focus very well. But as we go farther out, |
the ray begins to deviate from the focus, perhaps by falling short, and a ray |
striking near the top edge comes down and misses the focus by quite a wide |
margin. So, instead of getting a point image, we get a smear. This efect is called |
spherical œberration, because it is a property of the spherical surfaces we use in |
place of the ripght shape. This could be remedied, for any specifc obJect distance, |
by re-forming the shape of the lens surface, or perhaps by using several lenses |
arranged so that the aberrations of the individual lenses tend to cancel each other. |
Lenses have another fault: light of diferent colors has diferent speeds, or |
diferent indices of refraction, in the glass, and therefore the focal length of a |
given lens is diferent for diferent colors. 5o iŸ we image a white spot, the image |
will have colors, because when we focus for the red, the blue is out of focus, or |
vice versa. This property is called chrormatic aberrotion. |
'There are still other faults. If the object is of the axis, then the focus really |
1snˆt perfect any more, when it gets far enough of the axis. 'Phe easiest way to |
verify this is to focus a lens and then tilt it so that the rays are coming in at a |
large angle from the axis. hen the image that is formed will usually be quite |
crude, and there may be no place where it focuses well. There are thus several |
kinds of errors in lenses that the optical designer tries to remedy by using many |
lenses to compensate each other”s errors. |
How careful do we have to be to eliminate aberrations? Is it possible to make |
an absolutely perfect optical system? Suppose we had built an optical system |
that is supposed to bring light exactly to a point. Now, arguing from the poïnt |
of view of least time, can we fnd a condition on how perfect the system has to |
be? The system will have some kind oŸ an entrance opening for the light. IÝ we |
take the farthest ray from the axis that can come to the focus (ïf the system |
is perfect, of course), the times for all rays are exactly equal. But nothing is |
perfect, so the question is, how wrong can the time be for this ray and not be |
worth correcting any further? That depends on how perfectb we want to make |
the image. But suppose we want to make the image as perfect as it possibly can |
be made. 'Then, of course, our impression is that we have to arrange that every |
ray takes as nearly the same time as possible. But ¡% turns out that this is not |
true, that beyond a certain point we are trying to do something that is too ñne, |
because the theory of geometrical optics does not workl |
--- Trang 483 --- |
Remember that the principle of least time 1s not an accurate formulation, |
unlike the principle of conservation of energy or the principle of conservation |
of momentum. 'Phe principle of least time is only an approzimation, and 1§ |
1s interesting to know how much error can be allowed and still not make any |
apparent diference. The answer is that if we have arranged that between the |
maximal ray—the worst ray, the ray that is farthest out—and the central ray, the |
diferenee in time is less than about the period that corresponds to one oscillation |
of the light, then there is no use improving it any further. Light is an oscillatory |
thing with a defñnite frequenecy that is related to the wavelength, and if we have |
arranged that the time diference for diferent rays is less than about a period, |
there is no use going any further. |
27-7 Resolving power |
Another interesting question—a very important technical question with all |
optical instruments—is how much resoluing pouer they have. IÝ we build a |
microscope, we want to see the objects that we are looking at. That means, for |
instance, that if we are looking at a bacterium with a spot on each end, we want |
to see that there are two dots when we magnify them. One might think that all |
we have to do is to get enough magnification——we can always add another lens, |
and we can always magnify again and again, and with the cleverness of designers, |
all the spherical aberrations and chromatic aberrations can be cancelled out, |
and there is no reason why we cannot keep on magnifying the image. So the |
limitations of a microscope are not that it is impossible to build a lens that |
magnifes more than 2000 diameters. We can build a system of lenses that |
magnifes 10,000 diameters, but we s#Z/ could not see two points that are too |
close together because of the limitations of geometrical opties, because of the |
fact that least time is not precise. |
To discover the rule that determines how far apart® bwo points have to be |
so that at the image they appear as separate points can be stated in a very |
beautiful way associated with the time it takes for diferent rays. Suppose that |
we disregard the aberrations now, and imagine that for a particular point ? |
(Fig. 27-9) all the rays rom object to image 7' take exactly the same tỉme. (It |
is not true, because it is not a perfect system, but that is another problem.) |
NÑow take another nearby point, P, and ask whether its image will be distinct |
trom 7” In other words, whether we can make out the diference between them. |
Of course, according to geometrical optics, there should be two point images, |
--- Trang 484 --- |
frtZ| C————_ T |
ầm =ứY |
Fig. 27-9. The resolving power of an optical system. |
but what we see may be rather smeared and we may not be able to make out |
that there are two points. The condition that the second poïnt is focused ín a |
distinctly diferent place from the first one is that the two times for the extreme |
rays P'ST and PT on each side of the big opening of the lenses to go Írom |
one end to the other, must sœø‡ be equal from the two possible obJect points to |
a given image point. Why? Because, if the times were equal, of course both |
would ƒocus at the same point. So the times are not going to be equal. But by |
how much do they have to difer so that we can say that both do noø‡ come to a |
common fÍocus, so that we can distinguish the 6wo image points? 'The general rule |
for the resolution of any optical instrument is this: two diferent point sources |
can be resolved only if one source is focused at such a point that the times for |
the maximal rays from the other source to reach that point, as compared with |
its own true image point, difer by more than one period. It is necessary that the |
diference in time between the top ray and the bottom ray to the rong focus |
shall exceed a certain amount, namely, approximately the period of oscillation of |
the light: |
ta — tị > 1/1, (27.17) |
where 1 is the frequency of the light (number of oscillations per second; also speed |
divided by wavelength). IÝ the distance of separation of the two points is called |
D, and 1ƒ the opening angle of the lens is called Ø, then one can demonstrate |
that (27.17) is exactly equivalent to the statement that 2 must exceed À/nsin 0, |
where ø is the index of refraction at and À is the wavelength. 'Phe smallest |
things that we can see are therefore approximately the wavelength of light. A |
corresponding formula exists for telescopes, which tells us the smallest diference |
in angle bebween two stars that can just be distinguished.* |
* "The angle is about À/D, where D is the lens diameter. Can you see why? |
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