صور الصفحة
PDF
النشر الإلكتروني

which is equal to a surface of area 2pλ, all points of which are at the same distance from P. If ks is the effect at P of a small surface s placed at O, the effect of the first zone is 2kpλ, and the effect of the whole wave, as has been shown, is equal to that of half the first zone. The wave being plane, the amplitude at P is the same as at O. Calling that amplitude a, kpλ = a, and hence

[blocks in formation]

We must conclude that if a wave-front is split up into a number of small elements, we arrive at a correct result in the case of an extended plane wave of amplitude a, if we take the effect at a point P of a small surface s as regards amplitude to be as/pλ. The surface s is here supposed to be so small that the distances of its various points from P do not differ by more than a small fraction of the wave-length. The occurrence of p in the denominator can readily be understood, as the effect of an independent source on a point at a distance may be expected to be such that the intensity varies inversely as the square of the distance. If this be granted, it also follows that A must occur in the denominator, as the factor of a must be of the dimensions of a number, and of the three quantities s, p, à involving the unit of length s occurs in the numerator and p in the denominator.

W

R

It may now be shown that the value of k just obtained also gives correct results, when the wave-front is spherical. In Fig. 58 let waves diverge from a point Q and let it be required to calculate the effect at P from one of the wave-fronts WF. The only difference there can be between this problem and the previous one lies in the magnitude of the first zone, which must therefore be recalculated.

HO

[ocr errors]

Fig. 58.

P

Let RH be drawn at right angles to PQ and let QO=q; PO= p ; RH=f; HO=t.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

The effect of the first zone as regards amplitude is equal to 2ks/π, where s is the surface of the zone. Substituting s=#f2 and k = a/pλ, where a is the amplitude at O, the amplitude at P which is half the effect of the first zone is found to be aq/(p + q) and varies therefore, as it should do, inversely as the distance from Q.

Returning to the case of plane waves we obtain another important result by considering the phase of the resultant vibration. The phase at P due to the action of any zone has been shown above to be half way between the phases due to portions of the zone which are respectively nearest to and furthest from P. Applying this to the central zone, the phase of the resultant vibration at P, if calculated in the usual way, should differ from that at O by

[merged small][merged small][ocr errors][subsumed][merged small]

But we know that the optical distance from 0 to P is simply p, and hence the difference in phase is 2-p/A. It follows that if we want to obtain the phase correctly at P by means of Huygens' principle, we must everywhere subtract a quarter of a wave-length from the optical distance, or imagine the wave-front to be shifted forward through that distance.

It should be clearly understood what it is that has been proved. An extended wave-front has been divided into zones, and grouped together in such a way that the effect of the whole wave was found equal to that of half the central zone which lies close to the pole 0. The effect of a small surface s at a distant point P being expressed by ks, the result has shown that the possible variations of k depending on the angle between the normal to the wave-front and the radius vector, do not enter into the question at all. We conclude that those portions of the effect which might depend on it, are eliminated by interference. Similarly the result is independent of any possible effect of the direction of vibration.

The division of the wave-front into zones, drawn so that the distance of their successive edges from the point at which the amplitude of light is to be estimated, increases by half a wave-length, has rendered it possible to apply Huygens' principle in a simple and effective way. This mode of treating the propagation of waves being due to Fresnel, the zones should be called "Fresnel Zones."

47. Laminar zones. Instead of dividing the wave-front into circular zones, it is often more convenient to perform the division in a different manner. Let P (Fig. 59) be the point at which the light is to be estimated and WF the wave-front. Divide WF into a number of parallel strips at right angles to a central line HK. Let LM be

such a strip, which may again be subdivided into smaller areas, chosen to be of such magnitudes that the resultant phases of two successive elementary areas are in opposite directions. If the strip be indefinitely

[ocr errors]
[ocr errors]

h

M

F

K

P

Fig. 59.

extended in both directions, we may form a series as in the previous article, and find in this way that the total effect must be some definite fraction of that element of LM which is nearest to the central line HK. The whole effect being proportional to the width of the strip t, we may put it equal to kht, where k is the factor previously determined, and h some linear quantity. This expression asserts nothing more than that the effect of the strip is equal to that of an area situated in the central line HK, having a width t and a height h. The same reasoning may be applied to each of the strips which are parallel to LM, and we finally reduce the effect of the wave-front to that of a horizontal strip of width h. This may once more be subdivided. As the strip of width t produces an effect at P equal to kht, the effect of a strip of width h must be kh2. Hence the effect of the complete wave-front is reduced to that of an area h2 placed at 0, O being the pole of P. If the amplitude is a, kh2 = a. Hence

[blocks in formation]

p being the distance OP; the effect as regards amplitude of a strip such as LM of width t is therefore ta/√pλ.

To obtain the resultant phase due to each strip, we make use of the previously established fact that in applying Huygens' principle, we obtain the optical distance by taking away a quarter of a wave-length from the actual distance between the source and the point at which the amplitude is required. We imagine therefore the whole wavefront to be brought nearer through that distance. Now the process of attaining the final resultant from the rectangular strips consists of two exactly equal steps, the first in obtaining the intermediate resultant of each vertical strip such as LM, and the second in summing up for the horizontal strip HK which represents that intermediate resultant. If the total effect of the two steps as regards phase, is to bring back the wave-front to its proper position, each step must contribute equally, and therefore the optical distance of each strip is obtained by taking away A/8 from the actual distance. When the wave-front is divided into strips, it follows therefore that for the calculation of phases, we must imagine each strip to be brought nearer by λ/8. Or for simplicity of calculation we may say that we may take the optical distance of a strip to be equal to its actual distance, if we correct the final result

by subtracting 1/8 from the calculated optical distance or 45° from the calculated phase.

W

M

We may now determine the widths, t, of the strips, so that their

[blocks in formation]

all the vertical strips has been

λ

8

p+ instead of p, and if the resultant phases of successive strips are in opposite directions, the optical distance of the centre of each must be

[blocks in formation]

This gives for the edges of the strips the equation

[blocks in formation]

The central strip wants special consideration. It would not be correct to say that its resultant phase is the arithmetic mean between that due to the vibrations at O and T1, its nearest and furthest points, because the distance from Q to the line HK passes through a minimum at O. Hence the phase at Q due to any vertical subdivision of the strip, does not alter uniformly with the distance of that subdivision from 0. It is found, however, that the error introduced by making the supposition for the second strip is already very small, and hence the above subdivision will give sufficiently nearly the dividing lines between the zones which yield alternately opposite phases at P, because if the sum of all the strips above the first gives a phase equal to that of the resultant of all the strips, including the first, the phase of the first strip must be opposite to that of the sum of the remaining ones, which is equal to that of the second strip.

We take in accordance with this argument

[blocks in formation]

showing that for nearly all purposes, the error introduced by the simplification we have made is negligible.

The width of successive strips is obtained from

[blocks in formation]

where X2 is neglected compared to pλ. Hence for the first strip

for the second strip

and generally

[ocr errors][subsumed]

tn=OT-OT-1

= {√ pλ {√4n-1-√4n — 5}.

The effect of the nth strip t, is, as regards amplitude :

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

The numerical values of the effects are given in Table III. for n = 2 to n = 12. They have been calculated from the above expression, except for the first strip, for which the method fails to give correct results. The effect of this strip may be obtained by calculating the numerical value to which the series approaches, leaving out the first strip.

The series to be summed up is:

— [(√7 – √3) − (√T1 − √7) + (√15 – √11)...............].

Its value is found to be 1725, and the effect of all strips on one side of O being 5, if the amplitude of the incident wave is unity, it follows that the first strip produces an effect equal to 6725, as it is in the opposite direction as regards phase to the resultant effect of the rest of the wave-front.

[blocks in formation]
« السابقةمتابعة »