صور الصفحة
PDF
النشر الإلكتروني

48. Preliminary discussion of problems in diffraction. When an obstacle is placed in the path of a wave-front and the shadow of the obstacle received on a screen, the boundary of the shadow is not sharp, but the light encroaches to some extent on the dark portions, while there are bright and dark fringes on the side towards the light. If we draw straight lines which proceeding from the source of light touch the shadow-throwing body, the intersections of these lines with the screen enclose what may be called the geometrical shadow, meaning thereby the shadow constructed according to the laws of geometrical optics. Owing to the fact that light consists of waves, the laws of geometrical optics are not strictly true, but the waves spread round the obstacle and encroach to some extent on the geometrical shadow. That they do not do so to a greater extent, was the principal difficulty of the wave theory in its earlier form. This bending round of the waves has been called the "Diffraction" of light. The simplest problems of Diffraction are those in which we imagine a plane or spherical wave to impinge on a plane perforated screen. Whatever

H'

[ocr errors]

K'

H

Δι

K

Fig. 61.

form or position the apertures HK, H'K' (Fig. 61) have, we can find the disturbP ance at a point P by Huygens' principle, if we know the disturbance at all points. of the openings. In the usual solutions of the problems, the assumption is made that the disturbance is the same at all points in the plane of the screen as it would be if the screen were away. In other words, the screen simply obstructs the light which falls on its opaque portions, but does not otherwise alter the motion of the medium. That the assumption is one which needs justification may be understood by contemplating e.g. the flow of water through a pipe, in which the stream lines are parallel straight lines, and imagining that at some place a diaphragm is introduced across the pipe, leaving only an aperture much narrower than its cross section. We should here obviously arrive at erroneous results if we were to assume that the velocity of the water at all points of the opening has not been altered by the introduction of the diaphragm. In the case of the ordinary diffraction effects, it is found that the results arrived at by the simplified calculation are in agreement with experiment. This is a consequence of the small size of the length of a wave of light as compared with the other linear magnitudes which enter into the calculation, the errors introduced being sensible only within a few wave-lengths of the obstacle.

We are allowed therefore to use Huygens' principle in its simple form, provided we correctly introduce the contribution which each small surface element s at a point S of the opening contributes to the amplitude at P. If r be the distance PS, the angle between r and

the perpendicular to the wave-front at S, and the angle between r and the direction of vibration, the effect for homogeneous vibration of a small surface s at P is according to Stokes:

s (1+ cos ) sin

2rλ

This expression is based on the assumption that the displacements in the openings are everywhere the same as if the screen were away. Lord Rayleigh, on the other hand, has shown that if the forces acting across the plane of the screen are the same as if the screen were absent, the effect of s would be

[merged small][ocr errors]

and has also pointed out that so far as the treatment of diffraction problems is concerned, the terms depending on and disappear in consequence of interference, so that we may with equal justice adopt the simpler expression arrived at in the previous article, and take the effect of an element at S to be according to convenience either sλr, or s/λp, where p is the shortest distance from P to the wave-front.

49. Babinet's principle. Two screens may be called complementary when the openings of one correspond exactly to the opaque portions of the other and vice versa. If b be the amplitude at P in the absence of any screen, and a1, a2 are vectors representing the vibration at P when either one or the other of two complementary screens is interposed, then the sum of the vectors a1 and a, is obviously equal to b.

The principle due to Babinet allows us, whenever we have calculated the effect of one screen, to obtain that of the complementary screen without further trouble. A little care is necessary in using the principle, to take correct account of the difference in phase. But one simple result may at once be deduced from it. If a, is zero, a must be equal to b. Hence at every point where there is no light with one of the screens, the intensity when the complementary screen is introduced, is equal to that observed when the light is unobstructed. This statement cannot however be reversed. If a, b, a1 may have any value between zero and 2a. This is made obvious by the diagram

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

=

(Fig. 62) in which OA represents the amplitude (b) of the unobstructed light; OB the equal amplitude (a) observed when one of the screens is introduced. BA is then that vector which together with OB has OA as resultant. If the point traces out the circle of radius a2, the vector BA changes in magnitude from zero to 202.

Let a

50. Shadows of a straight edge in parallel light. plane wave-front WF (Fig. 63) fall upon a screen ME having a straight vertical edge passing through E, the plane of the drawing being horizontal, and let it be required to find the distribution of light on a distant and parallel screen SS'. Draw the wave-front which passes through E, and divide up that portion EG of the wave-front which is not blocked out by the screen, into suitable zones; EP being the normal to the wave-front P lies on

[blocks in formation]

T. T2 T3

[ocr errors]

the edge of the geometrical shadow. G At P the active wave-front EG represents one of two exactly symmetrical halves of the complete wave-front, which would operate if the screen were away. Hence the introduction of the screen reduces the amplitude at the geometrical shadow to one half and the intensity to one quarter. To find the amplitude at inside the geometrical shadow, construct Fresnel zones

Q P
Fig. 63.

some point

such that

λ

2

S'

=T1Q-EQ=T2Q - T1Q= T2 Q - T2 Q = ·

Unless is close to P, the resultant vibration due to the different zones will be alternately in opposite directions, and calling the effects of successive zones m1, m2, etc. the total effect is

[subsumed][ocr errors][merged small]

In this case the values of m diminish too quickly to allow us to write down the sum asm. It will however be some fraction of m1, and as with increasing distances of Q from P, each of the zones diminishes in width, the effect at Q is the smaller the further that point lies inside the geometrical shadow. The intensity which as has been shown is only 25 that of the incident light at the edge of the geometrical shadow, rapidly diminishes still further towards the inside of the shadow and soon becomes inappreciable.

If the point lies outside the geometrical shadow the intensities are obtained by drawing the normal to the wave-front, and the Fresnel zones, according to Art. 47.

The total effect in amplitude of that portion of the wave-front which lies to the right of the pole, when the shadow-throwing edge is on the left, is equal to 5, and the effect of the portion included between the pole and the edge is a maximum or a minimum, according as an odd or even number of zones are included between O and E (Fig. 60). The first maximum takes place when Q is at such a distance from P that OE=OT1. If the amplitude of the incident light is unity, and

the effects of successive zones are m1, m2, etc. the first maximum has an amplitude 5+ m1, half the amplitude of the incident light being added to represent that complete part of the wave which lies to the right of O. When Q has a position such that OE=OT2, there is a minimum with an amplitude 5+ m1- m2. The next maximum has a value ·5 + m1 — m2 + m, and though the maxima and minima rapidly approach each other in magnitude the intensity continues to oscillate about its mean value as the point Q is moved away from the geometrical shadow. The distances (x) of the maxima and minima from the edge are obtained from

[blocks in formation]

The equation shows that the loci of the maxima and minima are parabolas.

TABLE IV.

Shadow of straight edge.

Distance of screen = 100, X= 5 × 10-5, amplitude of incident light=1.

[blocks in formation]

The angle a/p being proportional to √λ/p is a small quantity unless n is large. But for large values of n the introduction of the screen causes no appreciable change in the distribution of light. Hence the effect of the screen is confined to the neighbourhood of its geometrical shadow. Table IV. gives the intensities of light at the first seven

maxima and six minima outside the geometrical shadow, and the intensities inside at the same distances from the edge. To give an idea of the scale, the positions to which the intensities refer are given for the case in which the shadow is received on a screen one metre away from the linear edge of the shadow-throwing object and the wave-length of light is 5 × 10-5 cms. The meaning of the last two columns will be explained in Art. 51.

The table shows that at a distance of 2.5 mm. from the edge of the geometrical shadow the light inside the shadow has only an intensity equal to the 500th part of that of the incident light, but that outside the shadow, at the same distance, the maximum and minimum intensities still differ by about 20%, while the interval between the bright and dark bands is 1 mm. The light must of course be homogeneous if it is desired to see more than a few of the bands. The distribution of the intensity of light in the neighbourhood of a straight edge is plotted in Fig. 64 from the numbers given

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

by Fresnel. The dotted vertical line represents the edge of the geometrical shadow where the intensity is one quarter. The distance of the screen from the edge is one metre and the scale of abscissæ represents millimetres.

M

51. Shadow of a straight edge in divergent light. If L

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

(Fig. 65) represents the source of light, which we suppose to be a luminous line parallel to the edge E which throws the shadow, we may for simplicity take the beam to have a cylindrical wave-front with the luminous source as axis. The traces of the wave-front with the plane of the paper are circles. Drawing EG, the wave-front, passing through the edge, we may divide it into laminar Fresnel zones, OT1, TT2, etc. which satisfy the condition that the

« السابقةمتابعة »