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spaced evenly, then all the intensities would indeed be equal. But, as a matter |
of fact, although we have taken the simplest case, we could also have considered |
an array of øœ#rs of antennas, in which each member of the pair has a certain |
strength and some relative phase. In this case, it is possible to get intensities |
which are diferent in the diferent orders. A grating is often made with little |
“sawtooth” cuts instead of little symmetrical notches. By carefully arranging the, |
“sawteeth,” more light may be sent into one particular order of spectrum than |
Into the others. In a practical grating, we would like to have as mụch light as |
possible in one of the orders. This may seem a complicated point to bring in, |
but it is a very clever thing to do, because it makes the grating more useful. |
So far, we have taken the case where all the phases of the sources are equal. |
But we also have a formula for ô when the phases difer from one to the nex$ |
by an angle œ. 'Phat requires wiring up our antennas with a slight phase shift |
between each one. Can we do that with light? Yes, we can do it very easily, |
for suppose that there were a source of light at infnity, œ an angle such that |
the light is coming in at an angle địn, and let us say that we wish to discuss the |
scattered beam, which is leaving at an angle Øẹu¿ (Fig. 30-4). The Ø¿u¿ is the |
--- Trang 519 --- |
đsin sụt đsin Ôn |
Fig. 30-4. The path difference for rays scattered from adjacent rulings |
of a grating Is đsin đụ: — đsin Ôịn. |
same Ø as we have had before, but the Ø¡ạ is merely a means for arranging that |
the phase of each source is diferent: the light coming from the distant driving |
source first hits one scratch, then the next, then the next, and so on, with a phase |
shift rom one to the other, which, as we see, is œ = —2dsinØ¡„/À. Therefore |
we have the formula for a grating in which light both comes in and goes out at |
an angle: |
@ = 2#dsin Øsu¿/À — 2xdsin Øịn /À. (30.7) |
Let us try to fñnd out where we get strong intensity in these cireumstances. The |
condition for strong intensities 1s, of course, that ø should be a multiple of 2z. |
'There are several interesting points to be noted. |
One case of rather great interest ¡is that which corresponds to m = 0, where |
đ is less than À; in fact, this is the only solution. In this case we see that |
sin Øsụy = sinØ¡n, which means that the light comes out in the sœmne điccfion as |
the light which was exciting the grating. We might think that the light “goes |
right through.” No, it 1s đjƒeren# light that we are talking about. The light that |
goes right through is from the original source; what we are talking about is the |
new light uhách ¡s generated bụ scaltering. It turns out that the scattered light |
1s going in the same direction as the original light, in fact it can interfere with |
it—a feature which we will study later. |
There is another solution for this same case. For a given Øịn, Øou„¿ may be |
the supplemen# oŸ Ø¡n. 5o not only do we get a beam in the same direction as |
the incoming beam but also one in another direction, which, if we consider 1E |
carefully, is such that the angle oƒ ïncidence ¡s equal to the angle oƒ scattering. |
'This we call the refected beam. |
--- Trang 520 --- |
So we begin to understand the basic machinery of reflection: the light that |
comes in generates motions of the atoms in the refector, and the refector then |
regeneraftes a. n„eu œue, and one of the solutions for the direction of scattering, |
the on solution if the spacing of the scatterers is small compared with one |
wavelength, is that the angle at which the light comes out is equal to the angle |
at which it comes inl |
Next, we discuss the special case when đ —> 0. 'PThat is, we have just a solid |
plece of material, so to speak, but of Ññnite length. In addition, we want the phase |
shift from one scatterer to the next to go to zero. In other words, we put more |
and more antennas between the other ones, so that each of the phase differences |
1s getting smaller, but the number of antennas is increasing in such a way that |
the total phase diference, between one end of the line and the other, is constant. |
Let us see what happens to (30.3) iƒ we keep the diference in phase nộ from one |
end to the other constant (say nó = ®), letting the number go to infnity and |
the phase shift ý of each one go to zero. But now ở is so small that sin ộ = ở, |
and if we also recognize n2fo as T„, the maximum intensity at the center of the |
beam, we find |
T= 4l2sin? ›®/8Ẻ. (30.8) |
This limiting case is what is shown in EFig. 30-2. |
In such cireumstances we fnd the same general kind of a picture as for ñnite |
spacing with đ > À; all the side lobes are practically the same as before, but |
there are no higher-order maxima. lf the scatterers are all in phase, we get a |
mmaximum in the direction Øs„y = 0, and a minimum when the distance A is equal |
to À, just as for finite đ and nø. 5o we can even analyze a con#nuous distribution |
Of scatterers or oscillators, by using integrals instead of summing. |
L † —— |
! =E=—ˆ——— †}——” |
Fig. 30-5. The intensity pattern of a continuous line of oscillators has |
a single strong maxImum and many weak “side lobes.” |
--- Trang 521 --- |
As an example, suppose there were a long line of oscillators, with the charge |
oscillating along the direction of the line (Eig. 30-5). Erom such an array the |
greatest intensity is perpendicular to the line. 'There is a little bit of intensity up |
and down from the equatorial plane, but it is very slipht. With this result, we can |
handle a more complicated situation. Suppose we have a set of such lines, each |
producing a beam only in a plane perpendicular to the line. To find the intensity |
in various directions from a series of long wires, Instead of infinitesimal wires, |
1s the same problem as it was for infñnitesimal wires, so long as we are in the |
central plane perpendicular to the wires; we just add the contribution from each |
of the long wires. hat is why, although we actually analyzed only tiny antennas, |
we might as well have used a grating with long, narrow slots. Each of the long |
slots produces an efect only in its own direction, not up and down, but they are |
all set next to each other horizontally, so they produce interference that way. |
Thus we can build up more complicated situations by having various distri- |
butions of scatterers in lines, planes, or in space. The first thing we did was to |
consider scatterers in a line, and we have just extended the analysis to strips; we |
can work it out by just doïng the necessary summations, adding the contributions |
from the individual scatterers. The principle is always the same. |
30-3 Resolving power of a grating |
W© are now in a position to understand a number of interesting phenomena. |
For example, consider the use oŸ a grating for separating wavelengths. We noticed |
that the whole spectrum was spread out on the screen, so a grating can be used |
as an instrument for separating light into its diferent wavelengths. One of the |
interesting questions is: supposing that there were t©wo sources of slightly diferent |
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