Page:EB1911 - Volume 08.djvu/257

half of that of the first existing zone, and this is sensibly the same as if there were no obstruction.

When light passes through a small circular or annular aperture, the illumination at any point along the axis depends upon the precise relation between the aperture and the distance from it at which the point is taken. If, as in the last paragraph, we imagine a system of zones to be drawn commencing from the inner circular boundary of the aperture, the question turns upon the manner in which the series terminates at the outer boundary. If the aperture be such as to fit exactly an integral number of zones, the aggregate effect may be regarded as the half of those due to the first and last zones. If the number of zones be even, the action of the first and last zones are antagonistic, and there is complete darkness at the point. If on the other hand the number of zones be odd, the effects conspire; and the illumination (proportional to the square of the amplitude) is four times as great as if there were no obstruction at all.

The process of augmenting the resultant illumination at a particular point by stopping some of the secondary rays may be carried much further (Soret, Pogg. Ann., 1875, 156, p. 99). By the aid of photography it is easy to prepare a plate, transparent where the zones of odd order fall, and opaque where those of even order fall. Such a plate has the power of a condensing lens, and gives an illumination out of all proportion to what could be obtained without it. An even greater effect (fourfold) can be attained by providing that the stoppage of the light from the alternate zones is replaced by a phase-reversal without loss of amplitude. R. W. Wood (Phil. Mag., 1898, 45, p. 513) has succeeded in constructing zone plates upon this principle.

In such experiments the narrowness of the zones renders necessary a pretty close approximation to the geometrical conditions. Thus in the case of the circular disk, equidistant (r ) from the source of light and from the screen upon which the shadow is observed, the width of the first exterior zone is given by

dx＝λ(2r )/4(2x),

2x being the diameter of the disk. If 2r＝1000 cm., 2x＝1 cm., λ＝6 × 10⁻⁵ cm., then dx＝·0015 cm. Hence, in order that this zone may be perfectly formed, there should be no error in the circumference of the order of ·001 cm. (It is easy to see that the radius of the bright spot is of the same order of magnitude.) The experiment succeeds in a dark room of the length above mentioned, with a threepenny bit (supported by three threads) as obstacle, the origin of light being a small needle hole in a plate of tin, through which the sun’s rays shine horizontally after reflection from an external mirror. In the absence of a heliostat it is more convenient to obtain a point of light with the aid of a lens of short focus.

The amplitude of the light at any point in the axis, when plane waves are incident perpendicularly upon an annular aperture, is, as above,

cos k(at − r ₁) − cos k(at − r ₂)＝2 sin kat sin k(r₁ − r ₂),

r ₂, r ₁ being the distances of the outer and inner boundaries from the point in question. It is scarcely necessary to remark that in all such cases the calculation applies in the first instance to homogeneous light, and that, in accordance with Fourier’s theorem, each homogeneous component of a mixture may be treated separately. When the original light is white, the presence of some components and the absence of others will usually give rise to coloured effects, variable with the precise circumstances of the case.

Although the matter can be fully treated only upon the basis of a dynamical theory, it is proper to point out at once that there is an element of assumption in the application of Huygens’s principle to the calculation of the effects produced by opaque screens of limited extent. Properly applied, the principle could not fail; but, as may readily be proved in the case of sonorous waves, it is not in strictness sufficient to assume the expression for a secondary wave suitable when the primary wave is undisturbed, with mere limitation of the integration to the transparent parts of the screen. But, except perhaps in the case of very fine gratings, it is probable that the error thus caused is insignificant; for the incorrect estimation of the secondary waves will be limited to distances of a few wave-lengths only from the boundary of opaque and transparent parts.

    Fig. 2.

3. Fraunhofer’s Diffraction Phenomena.—A very general problem in diffraction is the investigation of the distribution of light over a screen upon which impinge divergent or convergent spherical waves after passage through various diffracting apertures. When the waves are convergent and the recipient screen is placed so as to contain the centre of convergency—the image of the original radiant point, the calculation assumes a less complicated form. This class of phenomena was investigated by J. von Fraunhofer (upon principles laid down by Fresnel), and are sometimes called after his name. We may conveniently commence with them on account of their simplicity and great importance in respect to the theory of optical instruments.

If ƒ be the radius of the spherical wave at the place of resolution, where the vibration is represented by cos kat, then at any point M (fig. 2) in the recipient screen the vibration due to an element dS of the wave-front is (§ 2)

$-{\frac {d\mathrm {S} }{\lambda \rho }}\sin k(at-\rho ),$

ρ being the distance between M and the element dS.

Taking co-ordinates in the plane of the screen with the centre of the wave as origin, let us represent M by ξ, η, and P (where dS is situated) by x, y, z. Then

ρ²＝(x − ξ)² + (y − η)² + z², ƒ²＝x² + y² + z²;

so that

ρ²＝ƒ² − 2xξ − 2yη + ξ² + η².

In the applications with which we are concerned, ξ, η are very small quantities; and we may take

$\rho =f\left\{1-{\frac {x\xi +y\eta }{f^{2}}}\right\}.$

At the same time dS may be identified with dxdy, and in the denominator ρ may be treated as constant and equal to ƒ. Thus the expression for the vibration at M becomes

$-{\frac {1}{\lambda f}}\iint \sin k\left\{at-f+{\frac {x\xi +y\eta }{f}}\right\}dxdy$

(1);

and for the intensity, represented by the square of the amplitude,

	${\big .}\mathrm {I} ^{2}={\frac {1}{\lambda ^{2}f^{2}}}\left[\iint \sin k{\frac {x\xi +y\eta }{f}}dxdy\right]^{2}$
	$+{\frac {1}{\lambda ^{2}f^{2}}}\left[\iint \cos k{\frac {x\xi +y\eta }{f}}dxdy\right]^{2}$	(2).

This expression for the intensity becomes rigorously applicable when ƒ is indefinitely great, so that ordinary optical aberration disappears. The incident waves are thus plane, and are limited to a plane aperture coincident with a wave-front. The integrals are then properly functions of the direction in which the light is to be estimated.

In experiment under ordinary circumstances it makes no difference whether the collecting lens is in front of or behind the diffracting aperture. It is usually most convenient to employ a telescope focused upon the radiant point, and to place the diffracting apertures immediately in front of the object-glass. What is seen through the eye-piece in any case is the same as would be depicted upon a screen in the focal plane.

Before proceeding to special cases it may be well to call attention to some general properties of the solution expressed by (2) (see Bridge, Phil. Mag., 1858).

If when the aperture is given, the wave-length (proportional to k⁻¹) varies, the composition of the integrals is unaltered, provided ξ and η are taken universely proportional to λ. A diminution of λ thus leads to a simple proportional shrinkage of the diffraction pattern, attended by an augmentation of brilliancy in proportion to λ⁻².

If the wave-length remains unchanged, similar effects are produced by an increase in the scale of the aperture. The linear dimension of the diffraction pattern is inversely as that of the aperture, and the brightness at corresponding points is as the square of the area of aperture.

If the aperture and wave-length increase in the same proportion, the size and shape of the diffraction pattern undergo no change.

We will now apply the integrals (2) to the case of a rectangular aperture of width a parallel to x and of width b parallel to y. The limits of integration for x may thus be taken to be −1/2a and +1/2a, and for y to be −1/2b, +1/2b. We readily find (with substitution for k of 2π/λ)

$\mathrm {I} ^{2}={\frac {a^{2}b^{2}}{f^{2}\lambda ^{2}}}\cdot {\frac {\sin ^{2}{\frac {\pi a\xi }{f\lambda }}}{\frac {\pi ^{2}a^{2}\xi ^{2}}{f^{2}\lambda ^{2}}}}\cdot {\frac {\sin ^{2}{\frac {\pi b\eta }{f\lambda }}}{\frac {\pi ^{2}b^{2}\eta ^{2}}{f^{2}\lambda ^{2}}}}$

(3),

as representing the distribution of light in the image of a mathematical point when the aperture is rectangular, as is often the case in spectroscopes.

The second and third factors of (3) being each of the form sin²u/u², we have to examine the character of this function. It vanishes when u＝mπ, m being any whole number other than zero. When u＝0, it takes the value unity. The maxima occur when

u＝tan u,

(4).

and then

sin2u / u2＝cos2u

(5).

To calculate the roots of (5) we may assume

u＝(m + 1/2)π − y＝U − y,