determines the distance p of the screen from the opening, the central fringe being bright when is odd and dark when n is even. When the point is not included in the geometrical beam of light which is bounded by the straight lines LB and LA, a similar reasoning leads to the conclusion that there is the centre of a bright or dark fringe at Q according as AQ-BQ is an odd or even multiple of half a wavelength.

55. Passage of light through a circular aperture. When the perforations in a screen are such that we can divide the screen into circular zones, the calculation of the intensities is very simple for points in the axis of the zones.

R

Fig. 72.

P

Let O (Fig. 72) be the centre of a small circular aperture in a screen, and OP a line at right angles to the screen which we shall call the axis. If it is required to determine the amplitude at P due to a wave-front of unit amplitude incident on the screen, which we shall consider in the first instance to be plane and parallel to it, we may divide the aperture into Fresnel's zones, which produce effects which are equal in magnitude but alternately opposite in direction. If the radius OR of the aperture is such that an even number of zones is included, the amplitude at P is zero; if an uneven number is included the amplitude is a maximum and equal to that due to the first zone, and therefore double that of the unobstructed wave. The introduction of the screen with small aperture doubles the amplitude therefore at certain points. The condition for maximum or minimum of light is if PO=p, OR=r,

where there is a maximum if n be odd and a minimum if n be even. The general expression for the amplitude on the axis is found by subdividing the aperture into a large number of small zones of equal areas. Their total effect, according to Art. 5, is (A sin a)/a where for a we must put half the difference in phase at P of the disturbances due respectively to the first and last zone, i.e. half the difference in phase corresponding to an optical length nλ. This gives:

A is the amplitude at P calculated on the supposition that the disturbances of all zones reach P in the same phase, which would according to Art. 46 be π2/pλ, i.e. the area of the aperture divided by γλ.

The amplitude at P is therefore 2 sin (π2/2pλ). The points of zero illumination which have already been determined are the nearer together the smaller the distance of P. Sideways from the axis, the amplitudes cannot be calculated by simple methods, but general considerations similar to those which lead to accurate results in the case of long rectangular openings, are sufficient to show that there must be rhythmical alternations in the illumination. Hence a screen placed across the axis will show bright and dark rings having at P a bright or dark centre according to the distance of P from the opening.

The case of a divergent beam of light presents no further difficulty. We may subdivide the spherical wave-front into zones of equal area and obtain again at P the amplitude

A sin a

α

B

Fig. 73.

q being the distance of L from the screen. A has the same value as before.

Hence the points of maximum and minimum illumination. are determined by

and the amplitude at the maximum is 2q/(p + q).

56. Shadow of a circular disc. OR (Fig. 74) being a circular

W

S

R

Lx

Fig. 74.

P

disc, a spherical wave-front diverging from L, a luminous point on the axis of the disc, will throw a shadow on a screen SS', the centre of the shadow being on the axis. If Fresnel zones are drawn on the wave-front, the total effect at P as regards amplitude may be determined as in Art. 46 to be the same as that due to half the first zone, and if the disc is small, the first zone surrounding the edge of the disc has the same area as the central zone at 0, which is covered by the disc. Hence the illumination at P is the same as if the disc were away. Round this central bright spot there are alternately dark and bright rings. It will be an interesting exercise for the student to deduce the constancy of illumination on the axis. of a shadow-throwing disc from Babinet's principle, making use of the amplitude at the bright and dark centres of the complementary circular aperture. The fact that the shadow of a circular disc has a bright spot at its centre was discovered experimentally in the early part of the 18th century, but had been forgotten again when about 100 years later Poisson deduced it as a consequence of the wave-theory of

light. Arago, who was unaware of the earlier experiment, tested Poisson's mathematical conclusion, and verified it.

57. Zone plates. On a plane screen draw with O as centre, circles which divide the Fresnel zones with respect to a point P on the normal OP, the wave-front being supposed to be plane. For the radii of the circle we have the relation

r2 = np.λ,

where Po is the distance OP, and where n takes the values 1, 2, 3 etc. for successive circles. Imagine the zones on the screen to be alternately opaque and transparent. Then if a wave-front proceeding in the direction PO falls on the screen, the phases due to all transparent zones are in agreement at P, and hence the amplitude at P will be Nm where m represents the effect of the first zone and N the total number of zones.

The amplitude at P will therefore be N times what it would be if the screen were away. Such a zone plate acts like a lens concentrating parallel light to a focus, the focal distance being po. If now the source of light is moved to a point q from the screen, the zones will again unite their effects at P provided (Art. 46)

The relation between object and image is therefore the same as for a lens.

Zone plates may be made by drawing circles on a sheet of paper, the radii of which are as the square roots of successive numbers, and painting the alternate zones in black. When a photograph on glass is taken of such a drawing, a plate is produced which satisfies the conditions of a zone plate. To prepare an effective zone plate involves great labour. Prof. R. W. Wood has published a reduced print of such a plate* from which other still more reduced copies may be prepared by photographic reproduction. Prof. Wood† has also described a photographic method by means of which zone plates may be made, which give for alternate zones a complete phase reversal. A more perfect imitation of a lens may thus be obtained.

58. Historical. Augustin Jean Fresnel was born on May 10th, 1788, in Normandy, and entered the Government service as an engineer. He was occupied with the construction of roads, but lost his position owing to his having joined a body of men who opposed

*Phil. Mag. XLV. p. 511. 1898.

+ Ibidem.

Napoleon's re-entry into France, after his escape from Elba. Reinstated after Waterloo, he remained some time living in a small village in Normandy where his first study of the phenomena of diffraction seems to have been made. Fresnel was always of weak health and died on July 14, 1827. The undulatory theory of Optics owes to Fresnel more than to any other single man. His earlier work on Interference had to a great extent been anticipated by Thomas Young, but he is undoubtedly the discoverer of the true explanation of Diffraction. Young had tried to explain the external fringes of a shadow by means of interference of the rays which passed near the shadow-throwing object and those that were reflected from its surface. Fresnel, starting with the same idea, soon found that it was wrong, and proved by conclusive experiments that the surface reflexion had nothing to do with the appearance of the fringes. He then showed by mathematical calculation that the limitation of the beam, by the shadow-throwing object, was alone sufficient to cause the rhythmic variations of intensity outside the shadow.

CHAPTER VI.

DIFFRACTION GRATINGS.

59. General theory of a grating. A grating is a surface having a periodical structure which impresses a periodical alteration of phase or intensity on a transmitted or reflected wave of light. The most common method of manufacturing a grating is to rule equidistant lines with a diamond point on a surface of glass or metal.

The diamond introduces a periodical structure, each portion of which is probably very irregular, but which is repeated at perfectly regular intervals, Fig. 75. If the grating is ruled on a plane surface, that surface is called the plane of the grating. Any plane passing through K corresponding points of the grooves such as A1, A2, A3, is parallel to the plane of the grating. We distinguish between "reflexion gratings" and "transmission gratings" according as they are ruled on an opaque surface, the reflected or scattered light being used, or a transparent plate, through which the light is transmitted.

C3

H

C1

Fig. 75.

Let a plane wave-front be incident parallel to the grating. Waves spread out from the different portions of the grooves which may be considered as centres of secondary disturbances. If the light be received on a distant screen, the resultant of all vibrations at each point may be determined. Consider that point of the screen which lies in a direction A, C2 from the grating, and draw a plane HK at right angles to that direction. As the optical distance from any point on HK to the corresponding point of the distant screen is the same, we may take the phases of the vibrations which are to be combined, to be the same as the phases at HK. We combine in the first place, those vibrations which are due to the secondary waves. coming from one of the grooves. Selecting any point on the groove A2, we may always express the phase of the resultant vibration due to the whole groove as that corresponding to an optical distance A, C, E,