Page:EB1911 - Volume 08.djvu/270

Accordingly,

φ = 1/2πσ2

(32);

and for the curvature,

dφ / dσ = πσ

(33).

Cornu remarks that this equation suffices to determine the general character of the curve. For the osculating circle at any point includes the whole of the curve which lies beyond; and the successive convolutions envelop one another without intersection.

Fig. 19.

The utility of the curve depends upon the fact that the elements of arc represent, in amplitude and phase, the component vibrations due to the corresponding portions of the primary wave-front. For by (30) dσ = dv, and by (2) dv is proportional to ds. Moreover by (2) and (31) the retardation of phase of the elementary vibration from PQ (fig. 17) is 2πδ/λ, or φ. Hence, in accordance with the rule for compounding vector quantities, the resultant vibration at B, due to any finite part of the primary wave, is represented in amplitude and phase by the chord joining the extremities of the corresponding arc (σ₂ − σ₁).

In applying the curve in special cases of diffraction to exhibit the effect at any point P (fig. 18) the centre of the curve O is to be considered to correspond to that point C of the primary wave-front which lies nearest to P. The operative part, or parts, of the curve are of course those which represent the unobstructed portions of the primary wave.

Let us reconsider, following Cornu, the diffraction of a screen unlimited on one side, and on the other terminated by a straight edge. On the illuminated side, at a distance from the shadow, the vibration is represented by JJ′. The co-ordinates of J, J′ being (1/2, 1/2), (−1/2, −1/2), I² is 2; and the phase is 1/8 period in arrear of that of the element at O. As the point under contemplation is supposed to approach the shadow, the vibration is represented by the chord drawn from J to a point on the other half of the curve, which travels inwards from J′ towards O. The amplitude is thus subject to fluctuations, which increase as the shadow is approached. At the point O the intensity is one-quarter of that of the entire wave, and after this point is passed, that is, when we have entered the geometrical shadow, the intensity falls off gradually to zero, without fluctuations. The whole progress of the phenomenon is thus exhibited to the eye in a very instructive manner.

We will next suppose that the light is transmitted by a slit, and inquire what is the effect of varying the width of the slit upon the illumination at the projection of its centre. Under these circumstances the arc to be considered is bisected at O, and its length is proportional to the width of the slit. It is easy to see that the length of the chord (which passes in all cases through O) increases to a maximum near the place where the phase-retardation is 3/8 of a period, then diminishes to a minimum when the retardation is about 7/8 of a period, and so on.

If the slit is of constant width and we require the illumination at various points on the screen behind it, we must regard the arc of the curve as of constant length. The intensity is then, as always, represented by the square of the length of the chord. If the slit be narrow, so that the arc is short, the intensity is constant over a wide range, and does not fall off to an important extent until the discrepancy of the extreme phases reaches about a quarter of a period.

We have hitherto supposed that the shadow of a diffracting obstacle is received upon a diffusing screen, or, which comes to nearly the same thing, is observed with an eye-piece. If the eye, provided if necessary with a perforated plate in order to reduce the aperture, be situated inside the shadow at a place where the illumination is still sensible, and be focused upon the diffracting edge, the light which it receives will appear to come from the neighbourhood of the edge, and will present the effect of a silver lining. This is doubtless the explanation of a “pretty optical phenomenon, seen in Switzerland, when the sun rises from behind distant trees standing on the summit of a mountain.”^[1]

11. Dynamical Theory of Diffraction.—The explanation of diffraction phenomena given by Fresnel and his followers is independent of special views as to the nature of the aether, at least in its main features; for in the absence of a more complete foundation it is impossible to treat rigorously the mode of action of a solid obstacle such as a screen. But, without entering upon matters of this kind, we may inquire in what manner a primary wave may be resolved into elementary secondary waves, and in particular as to the law of intensity and polarization in a secondary wave as dependent upon its direction of propagation, and upon the character as regards polarization of the primary wave. This question was treated by Stokes in his “Dynamical Theory of Diffraction” (Camb. Phil. Trans., 1849) on the basis of the elastic solid theory.

Let x, y, z be the co-ordinates of any particle of the medium in its natural state, and χ, η, ζ the displacements of the same particle at the end of time t, measured in the directions of the three axes respectively. Then the first of the equations of motion may be put under the form

${\frac {d^{2}\xi }{dt^{2}}}{=}b^{2}\left({\frac {d^{2}\xi }{dx^{2}}}+{\frac {d^{2}\xi }{dy^{2}}}+{\frac {d^{2}\xi }{dz^{2}}}\right)+(a^{2}-b^{2}){\frac {d}{dx}}\left({\frac {d\xi }{dx}}+{\frac {d\eta }{dy}}+{\frac {d\zeta }{dz}}\right),$

where a² and b² denote the two arbitrary constants. Put for shortness

${\frac {d\xi }{dx}}+{\frac {d\eta }{dy}}+{\frac {d\zeta }{dz}}=\delta$

(1),

and represent by ∇²ξ the quantity multiplied by b². According to this notation, the three equations of motion are

	${\frac {d^{2}\xi }{dt^{2}}}=b^{2}\nabla ^{2}\xi +(a^{2}-b^{2}){\frac {d\delta }{dx}}$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$	(2).
	${\frac {d^{2}\eta }{dt^{2}}}=b^{2}\nabla ^{2}\eta +(a^{2}-b^{2}){\frac {d\delta }{dy}}$
	${\frac {d^{2}\zeta }{dt^{2}}}=b^{2}\nabla ^{2}\zeta +(a^{2}-b^{2}){\frac {d\delta }{dz}}$

It is to be observed that S denotes the dilatation of volume of the element situated at (x, y, z). In the limiting case in which the medium is regarded as absolutely incompressible δ vanishes; but, in order that equations (2) may preserve their generality, we must suppose a at the same time to become infinite, and replace a²δ by a new function of the co-ordinates.

These equations simplify very much in their application to plane waves. If the ray be parallel to OX, and the direction of vibration parallel to OZ, we have ξ = 0, η = 0, while ζ is a function of x and t only. Equation (1) and the first pair of equations (2) are thus satisfied identically. The third equation gives

${\frac {d^{2}\zeta }{dt^{2}}}{=}b^{2}{\frac {d^{2}\zeta }{dx^{2}}}$

(3),

of which the solution is

ζ = ƒ(bt − x)

(4),

where ƒ is an arbitrary function.

The question as to the law of the secondary waves is thus answered by Stokes. “Let ξ = 0, η = 0, ζ = ƒ(bt − x) be the displacements corresponding to the incident light; let O₁ be any point in the plane P (of the wave-front), dS an element of that plane adjacent to O₁, and consider the disturbance due to that portion only of the incident disturbance which passes continually across dS. Let O be any point in the medium situated at a distance from the point O₁ which is large in comparison with the length of a wave; let O₁O = r, and let this line make an angle θ with the direction of propagation of the incident light, or the axis of x, and φ with the direction of vibration, or axis of z. Then the displacement at O will take place in a direction perpendicular to O₁O, and lying in the plane ZO₁O; and, if ζ′ be the displacement at O, reckoned positive in the direction nearest to that in which the incident vibrations are reckoned positive,

$\zeta ^{\prime }{=}{\frac {d\mathrm {S} }{4\pi r}}(1+\cos \theta )\sin \phi f^{\prime }(bt-r).$

In particular, if

$f(bt-x){=}c\sin {\frac {2\pi }{\lambda }}(bt-x)$

(5),

we shall have

$\zeta ^{\prime }{=}{\frac {cd\mathrm {S} }{2\lambda r}}(1+\cos \theta )\sin \phi \cos {\frac {2\pi }{\lambda }}(bt-r)$

(6).”

It is then verified that, after integration with respect to dS, (6) gives the same disturbance as if the primary wave had been supposed to pass on unbroken.

The occurrence of sin φ as a factor in (6) shows that the relative intensities of the primary light and of that diffracted in the direction θ depend upon the condition of the former as regards polarization. If the direction of primary vibration be perpendicular to the plane of diffraction (containing both primary and secondary rays), sin φ = 1; but, if the primary vibration be in the plane of diffraction, sin φ = cos θ. This result was employed by Stokes as a criterion of the direction of vibration; and his experiments, conducted with gratings, led him to the conclusion that the vibrations

1. H. Necker (Phil. Mag., November 1832); Fox Talbot (Phil. Mag., June 1833). “When the sun is about to emerge ... every branch and leaf is lighted up with a silvery lustre of indescribable beauty.... The birds, as Mr Necker very truly describes, appear like flying brilliant sparks.” Talbot ascribes the appearance to diffraction; and he recommends the use of a telescope.