Page:EB1911 - Volume 08.djvu/268

be regarded as formed of infinitely thin wires, or infinitely narrow lines traced upon glass. The secondary pulses diverted by the ruling fall upon an object-glass as usual, and on arrival at the focus constitute a procession equally spaced in time, the interval between consecutive members depending upon the obliquity. If a retarding plate be now inserted so as to operate upon the pulses which come from one side of the grating, while leaving the remainder unaffected, we have to consider what happens at the focal point chosen. A full discussion would call for the formal application of Fourier’s theorem, but some conclusions of importance are almost obvious.

Previously to the introduction of the plate we have an effect corresponding to wave-lengths closely grouped around the principal wave-length, viz. σ sin φ, where σ is the grating-interval and φ the obliquity, the closeness of the grouping increasing with the number of intervals. In addition to these wave-lengths there are other groups centred round the wave-lengths which are submultiples of the principal one—the overlapping spectra of the second and higher orders. Suppose now that the plate is introduced so as to cover naif the aperture and that it retards those pulses which would otherwise arrive first. The consequences must depend upon the amount of the retardation. As this increases from zero, the two processions which correspond to the two halves of the aperture begin to overlap, and the overlapping gradually increases until there is almost complete superposition. The stage upon which we will fix our attention is that where the one procession bisects the intervals between the other, so that a new simple procession is constituted, containing the same number of members as before the insertion of the plate, but now spaced at intervals only half as great. It is evident that the effect at the focal point is the obliteration of the first and other spectra of odd order, so that as regards the spectrum of the first order we may consider that the two beams interfere. The formation of black bands is thus explained, and it requires that the plate be introduced upon one particular side, and that the amount of the retardation be adjusted to a particular value. If the retardation be too little, the overlapping of the processions is incomplete, so that besides the procession of half period there are residues of the original processions of full period. The same thing occurs if the retardation be too great. If it exceed the double of the value necessary for black bands, there is again no overlapping and consequently no interference. If the plate be introduced upon the other side, so as to retard the procession originally in arrear, there is no overlapping, whatever may be the amount of retardation. In this way the principal features of the phenomenon are accounted for, and Schuster has shown further how to extend the results to spectra having their origin in prisms instead of gratings.

10. Diffraction when the Source of Light is not seen in Focus.—The phenomena to be considered under this head are of less importance than those investigated by Fraunhofer, and will be treated in less detail; but in view of their historical interest and of the ease with which many of the experiments may be tried, some account of their theory cannot be omitted. One or two examples have already attracted our attention when considering Fresnel’s zones, viz. the shadow of a circular disk and of a screen circularly perforated.

Fresnel commenced his researches with an examination of the fringes, external and internal, which accompany the shadow of a narrow opaque strip, such as a wire. As a source of light he used sunshine passing through a very small hole perforated in a metal plate, or condensed by a lens of short focus. In the absence of a heliostat the latter was the more convenient. Following, unknown to himself, in the footsteps of Young, he deduced the principle of interference from the circumstance that the darkness of the interior bands requires the co-operation of light from both sides of the obstacle. At first, too, he followed Young in the view that the exterior bands are the result of interference between the direct light and that reflected from the edge of the obstacle, but he soon discovered that the character of the edge—e.g. whether it was the cutting edge or the back of a razor—made no material difference, and was thus led to the conclusion that the explanation of these phenomena requires nothing more than the application of Huygens’s principle to the unobstructed parts of the wave. In observing the bands he received them at first upon a screen of finely ground glass, upon which a magnifying lens was focused; but it soon appeared that the ground glass could be dispensed with, the diffraction pattern being viewed in the same way as the image formed by the object-glass of a telescope is viewed through the eye-piece. This simplification was attended by a great saving of light, allowing measures to be taken such as would otherwise have presented great difficulties.

Fig. 17.

In theoretical investigations these problems are usually treated as of two dimensions only, everything being referred to the plane passing through the luminous point and perpendicular to the diffracting edges, supposed to be straight and parallel. In strictness this idea is appropriate only when the source is a luminous line, emitting cylindrical waves, such as might be obtained from a luminous point with the aid of a cylindrical lens. When, in order to apply Huygens’s principle, the wave is supposed to be broken up, the phase is the same at every element of the surface of resolution which lies upon a line perpendicular to the plane of reference, and thus the effect of the whole line, or rather infinitesimal strip, is related in a constant manner to that of the element which lies in the plane of reference, and may be considered to be represented thereby. The same method of representation is applicable to spherical waves, issuing from a point, if the radius of curvature be large; for, although there is variation of phase along the length of the infinitesimal strip, the whole effect depends practically upon that of the central parts where the phase is sensibly constant. ^[1]

In fig. 17 APQ is the arc of the circle representative of the wave-front of resolution, the centre being at O, and the radius QA being equal to a. B is the point at which the effect is required, distant a + b from O, so that AB = b, AP = s, PQ = ds.

Taking as the standard phase that of the secondary wave from A, we may represent the effect of PQ by

$\cos 2\pi \left({\frac {t}{\tau }}-{\frac {\delta }{\lambda }}\right)\cdot ds,$

where δ = BP − AP is the retardation at B of the wave from P relatively to that from A. Now

δ＝(a + b) s2/2ab

(1),

so that, if we write

${\frac {2\pi \delta }{\lambda }}{=}{\frac {\pi (a+b)s^{2}}{ab\lambda }}{=}{\frac {\pi }{2}}v^{2}$

(2),

the effect at B is

$\left\{{\frac {ab\lambda }{2(a+b)}}\right\}^{\frac {1}{2}}\left\{\cos {\frac {2\pi t}{\tau }}\int \cos {\tfrac {1}{2}}\pi v^{2}.dv+\sin {\frac {2\pi t}{\tau }}\int \sin {\tfrac {1}{2}}\pi v^{2}\cdot dv\right\}$

(3)

the limits of integration depending upon the disposition of the diffracting edges. When a, b, λ are regarded as constant, the first factor may be omitted,—as indeed should be done for consistency’s sake, inasmuch as other factors of the same nature have been omitted already.

The intensity I², the quantity with which we are principally concerned, may thus be expressed

I2 = { ∫ cos 1/2πv2·dv}2 + { ∫ sin 1/2πv2·dv }2

(4).

These integrals, taken from v = 0, are known as Fresnel’s integrals; we will denote them by C and S, so that

$\mathrm {C} =\int _{0}^{v}\cos {\frac {1}{2}}\pi v^{2}\cdot dv,\qquad \qquad \mathrm {S} =\int _{0}^{v}\sin {\frac {1}{2}}\pi v^{2}\cdot dv$

(5).

When the upper limit is infinity, so that the limits correspond to the inclusion of half the primary wave, C and S are both equal to 1/2, by a known formula; and on account of the rapid fluctuation of sign the parts of the range beyond very moderate values of v contribute but little to the result.

Ascending series for C and S were given by K. W. Knockenhauer, and are readily investigated. Integrating by parts, we find

$\mathrm {C} +i\mathrm {S} =\int _{0}^{v}e^{i.{\frac {1}{2}}\pi v^{2}}dv=e^{i.{\frac {1}{2}}\pi v^{2}}.v-{\tfrac {1}{3}}i\pi \int _{0}^{v}e^{i.{\frac {1}{2}}\pi v^{2}}dv^{3};$

and, by continuing this process,

$\mathrm {C} +i\mathrm {S} =e^{i.{\frac {1}{2}}\pi v^{2}}\left\{v-{\frac {i\pi }{3}}v^{3}+{\frac {i\pi }{3}}{\frac {i\pi }{5}}v^{5}-{\frac {i\pi }{3}}{\frac {i\pi }{5}}{\frac {i\pi }{7}}v^{7}+\ldots \right\}.$

By separation of real and imaginary parts,

	$\mathrm {C} =\mathrm {M} \cos {\tfrac {1}{2}}\pi v^{2}+\mathrm {N} \sin {\tfrac {1}{2}}\pi v^{2}$		$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$	(6),
	$\mathrm {S} =\mathrm {M} \sin {\tfrac {1}{2}}\pi v^{2}-\mathrm {N} \cos {\tfrac {1}{2}}\pi v^{2}$

where

$\mathrm {M} ={\frac {v}{1}}-{\frac {\pi ^{2}v^{5}}{3.5}}+{\frac {\pi ^{4}v^{9}}{3.5.7.9}}-\ldots$

(7),

$\mathrm {N} ={\frac {\pi v^{3}}{1.3}}-{\frac {\pi ^{3}v^{7}}{1.3.5.7}}+{\frac {\pi ^{5}v^{11}}{1.3.5.7.9.11}}\ldots$

(8).

These series are convergent for all values of v, but are practically useful only when v is small.

Expressions suitable for discussion when v is large were obtained

1. In experiment a line of light is sometimes substituted for a point in order to increase the illumination. The various parts of the line are here independent sources, and should be treated accordingly. To assume a cylindrical form of primary wave would be justifiable only when there is synchronism among the secondary waves issuing from the various centres.