text stringlengths 0 6.73k |
|---|
inñnity focuses. Once we had the focal length, it would be better to write our |
equation in terms of the focal length directly, and the formula then is |
(1/s) + (1/5) = 1/. (27.12) |
Now let us see how the formula works and what it implies in diferent circum- |
siances. First, it implies that IÝ s or sf is inñnite the other one is ƒ. That means |
that parallel light focuses at a distance ƒ, and this in efect defines ƒ. Another |
interesting thing i% says is that both points move in the same direction. lf one |
moves to the right, the other does also. Another thing it says is that s and s/ |
are equal if they are both equal to 2ƒ. In other words, if we want a symmetrical |
situation, we ñnd that they will both focus at a distance 2ƒ. |
--- Trang 479 --- |
27-4 Magnification |
So far we have discussed the focusing action only for points on the axis. NÑow |
let us discuss also the imaging of objects not exactly on the axis, but a little |
bít of, so that we can understand the properties of mmagnification. When we set |
up a lens so as to focus light from a smaill filament onto a “point” on a screen, |
we notice that on the screen we get a “picture” of the same filament, except of |
a larger or smaller size than the true fñlament. 'Phis must mean that the light |
comes to a focus from cach poøzn‡ of the filament. In order to understand this a |
little better, let us analyze the thin lens system shown schematically in Eig. 27-7. |
W©e know the following facts: |
(1) Any ray that comes in parallel on one side proceeds toward a certain |
particular point called the focus on the other side, at a distance ƒ from the |
(2) Any ray that arrives at the lens from the focus on one side comes out |
parallel to the axis on the other side. |
Thịs is all we need to establish formula (27.12) by geometry, as follows: Suppose |
we have an object at some distance #ø from the focus; let the height of the object |
be . Then we know that one of the rays, namely PQ, will be bent so as to pass |
through the focus # on the other side. Now ïf the lens will focus point P at all, |
we can fnd out where if we fnd out where just one other ray goes, because the |
new focus will be where the two intersect again. We need only use our ingenulty |
to fñnd the exact direction of øne other ray. But we remember that a parallel |
ray goes through the focus and 0e 0ersa: a ray which goes through the focus |
will come out parallel' So we draw ray 7 through . (It is true that the actual |
rays which are doing the focusing may be much more limited than the two we |
have drawn, but they are harder to fñgure, so we make believe that we can make |
ÔN CƯ ƯỢNG cu |
Fig. 27-7. The geometry of imaging by a thin lens. |
--- Trang 480 --- |
this ray.) Since it would come out parallel, we draw 79 parallel to XW. The |
Intersection ®Š is the point we need. 'Phis will determine the correcE place and |
the correct height. Let us call the height ˆ and the distance from the focus, z. |
Now we may derive a lens formula. Ủsing the similar triangles PVU and 7'XU, |
we fnd , |
—==~. 27.13 |
Th (713) |
Similarly, from triangles SW . and QX, we get |
Ụ _— 9 |
—==_.. 27.14 |
n=Ủ (2714) |
Solving each for /, we ñnd that |
U_—# (27.15) |
Equation (27.15) is the famous lens formula; in it is everything we need to know |
about lenses: Ib tells us the magnification, #“/, in terms of the distances and |
the focal lengths. It also connects the bwo distances z and øˆ with ƒ: |
ma! = Ƒ, (27.16) |
which is a much neater form to work with than Eq. (27.12). We leave it to the |
student to demonstrate that if we call s = #z + ƒ and s” = zø' + ƒ, Bq. (27.12) is |
the same as Eq. (27.16). |
27-5 Compound lenses |
Without actually deriving it, we shall briefy describe the general result when |
we have a number of lenses. If we have a system of several lenses, how can |
we possibly analyze it? 'Phat is easy. We start with some object and calculate |
where its image is for the first lens, using formula (27.16) or (27.12) or any other |
equivalent formula, or by drawing diagrams. So we fñnd an image. Then we treat |
this image as the source for the next lens, and use the second lens with whatever |
1ts focal length is to again ñnd an image. We simply chase the thing through |
the succession of lenses. 'That is all there is to it. It involves nothing new in |
principle, so we shall not go into it. However, there is a very interesting net |
--- Trang 481 --- |
result of the efects of any sequence of lenses on light that starts and ends up in |
the same medium, say air. Any optical instrument——a telescope or a microscope |
with any number of lenses and mirrors—has the following property: There exist |
two planes, called the prinecipal pÏøœnes of the system (these planes are often fairly |
close to the first surface of the first lens and the last surface of the last lens), |
which have the following properties: (1) If light comes into the system parallel |
from the first side, it comes out at a certain focus, at a distance from the second |
principal plane equal to the focal length, Just as though the system were a thin |
lens situated at this plane. (2) Tf parallel light comes in the other way, i1 comes |
to a focus at the same distance ƒ from the ƒrs‡ principal plane, again as If a thin |
lens where situated there. (See Eig. 27-8.) |
Fig. 27-8. lllustration of the principal planes of an optical system. |
Of course, iŸ we measure the distances # and z', and ÿ and z as before, |
the formula (27.16) that we have written for the thin lens is absolutely general, |
provided that we measure the focal length from the principal planes and not from |
the center of the lens. It so happens that for a thin lens the principal planes are |
coincident. It is just as though we could take a thin lens, slice i2 down the middle, |
and separate it, and not notice that it was separated. Every ray that comes in |
pops out immediately on the other side of the second plane from the same point |
as it went into the first planel 'The principal planes and the focal length may be |
found either by experiment or by calculation, and then the whole set oŸ propertfies |
of the optical system are described. lt is very Interesting that the result is not |
complicated when we are all ñnished with such a big, complicated optical system. |
27-6 Aberrations |
Before we get too excited about how marvelous lenses are, we must hasten |
to add that there are also serious limitations, because of the fact that we have |
--- Trang 482 --- |
limited ourselves, strictly speaking, to paraxial rays, the rays near the axis. A |
real lens having a fñnite size will, in general, exhibit aberrations. For example, |
a ray that is on the axis, of course, goes through the focus; a ray that is very |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.