DIFFRACTION OF LIGHT.—i. When light proceeding from a small source falls upon an opaque object, a shadow is cast upon a screen situated behind the obstacle, and this shadow is found to be bordered by alternations of brightness and darkness, known as " diffraction bands." The phenomena thus presented were described by Grimaldi and by Newton. Subsequently T. Young showed that in their formation interference plays an important part, but the complete explanation was reserved for A. J. FresneI. Later investigations by Fraunhofer, Airy and others have greatly widened the field, and under the head of " diffraction " are now usually treated all the effects dependent upon the limitation of a beam. of light, as well as those which arise from irregularities of. any kind at surfaces through which it is transmitted, or at which it is reflected.
2. Shadows.—In the infancy of the undulatory theory the objection most frequently urged against it was the difficulty of explaining the very existence of shadows. Thanks to Fresnel and his followers, this department of optics is now precisely the one in which the theory has gained its greatest triumphs. The principle employed in these investigations is due to C. Huygens, and may be thus formulated. If round the origin of waves an ideal closed surface be drawn, the whole action of the waves in the region beyond may be regarded as due to the motion continually propagated across the various elements of this surface. The wave motion due to any element of the surface is called a secondary wave, and in estimating the total effect regard must be paid to the phases as well as the amplitudes of the components. It is usually convenient to choose as the surface of resolution a wave front, i.e. a surface at which the primary vibrations are in one phase. Any obscurity that may hang over Huygens's principle is due mainly to the indefiniteness of thought and expression which we must be content to put up with if we wish to avoid pledging ourselves as to the character of the vibrations. In the application to sound, where we know what we are dealing with, the matter is simple enough in principle, although mathematical difficulties would often stand in the way of the calculations we might wish to make.
By this expression, in conjunction with the quarterperiod acceleration of phase, the law of the secondary wave is determined. That the amplitude of the secondary wave should vary as r' was to be expected from considerations respecting energy; but the occurrence of the factor a', and the acceleration of phase, have sometimes been regarded as mysterious. It may be well therefore to remember that precisely these laws apply to a secondary wave of sound, which can be investigated upon the strictest mechanical principles.
The recomposition of the secondary waves may also be treated analytically. If the primary wave at 0 be cos kat, the effect of the secondary wave proceeding from the element dS at Q is
dS 1 dS
p cos k(atp+)) = gip sin k(atp).
If dS=2axdx, we have for the whole effect 2a ('m sink(atp)xdx
f
X o p
The ideal surface of resolution may be there regarded as a flexible lamina; and we know that, if by forces locally applied every element of the lamina be made to move normally to itself exactly as the air at that place does, the external aerial motion is fully determined. By the principle of superposition the whole effect may be found by integration of the partial effects due to each element of the surface, the other elements remaining at rest.
We will now consider in detail the important case in which uniform plane waves are resolved at a surface coincident with a wavefront
(OQ). We imagine a wavefront divided
6 x into elementary rings or zones—often named
after Huygens, but better after Fresnel
by spheres described round P (the point at
which the aggregate effect is to be estimated),
the first sphere, touching the plane at 0, with
Y ey a radius equal to P0, and the succeeding spheres with radii increasing at each step by 4X. There are thus marked out a series of circles, whose radii x are given by x2+r2=(r+anX)2, or x2=nar nearly; so that the rings are at first of nearly equal area. FIG. 1. Now the effect upon P of each element of the
plane is proportional to its area; but it depends also upon the distance from P, and possibly upon the inclination of the secondary ray to the direction of vibration and to the wavefront.
The latter question can only be treated in connexion with the dynamical theory (see below, § i I); but under all ordinary circumstances the result is independent of the precise answer that may be given. All that it is necessary to assume is that the effects of the successive zones gradually diminish, whether from the increasing obliquity of the secondary ray or because (on account of the limitation of the region of integration) the zones become at last more and more incomplete. The component vibrations at P due to the successive zones are thus nearly equal in amplitude. and opposite in phase (the phase of each corresponding to that of the infinitesimal circle midway between the boundaries), and the series which we have to sum is one in which the terms are alternately opposite in sign and, while at first nearly constant in numerical magnitude, gradually diminish to zero. In such a series each term may be regarded as very nearly indeed destroyed by the halves of its immediate neighbours, and thus the sum of the whole series is represented by half the first term, which stands over uncompensated. The question is thus reduced to that of finding the effect of the first zone, or central circle, of which the area is vrXr.
We have seen that the problem before us is independent of the law of the secondary wave as regards obliquity; but the result of the integration necessarily involves the law of the intensity and phase of a secondary wave as a function of r, the distance from the origin. And we may in fact, as was done by A. Smith (Camb. Math. Journ., 1843, 3, p. 46), determine the law of the secondary wave, by comparing the result of the integration with that obtained by supposing the primary wave to pass on to P without resolution.
Now as to the phase of the secondary wave, it might appear natural to suppose that it starts from any point Q with the phase of the primary wave, so that on arrival at P, it is retarded by the amount corresponding to QP. But a little consideration will prove that in that case the series of secondary waves could not reconstitute the primary wave. For the aggregate effect of the secondary waves is the half of that of the first Fresnel zone, and it is the central element only of that zone for which the distance to be travelled is equal to r. Let us conceive the zone in question to be divided into infinitesimal rings of equal area. The effects due to each of these rings are equal in amplitude and of phase ranging uniformly over half a complete period. The phase of the resultant is midway between those of the extreme elements, that is to say, a quarter of a period behind that due to the element at the centre of the circle. It is accordingly necessary to suppose that the secondary waves start with a phase onequarter of a period in advance of that of the primary wave at the surface of resolution.
Further, it is evident that account must be taken of the variation of phase in estimating the magnitude of the effect at P of the first zone. The middle element alone contributes without deduction; the effect of every other must be found by introduction of a resolving factor, equal to cos 0, if 0 represent the difference of phase between this element and the resultant. Accordingly, the amplitude of the resultant will be less than if all its components had the same phase, in the ratio
I cos Bd6 : a,
/ ar
or 2: u. Now 2 area /ar=2Xr; so that, in order to reconcile the amplitude of the primary wave (taken as unity) with the half effect of the first zone, the amplitude, at distance r, of the secondary wave emitted from the element of area dS must be taken to be
dS/Xr (1).or, since xdx = pdp, k =2a/a,
sin k(atp)d4= [cos k(atp)];
In order to obtain the effect of the primary wave, as retarded by traversing the distance r, viz. cos k(atr), it is necessary to suppose that the integrated term vanishes at the upper limit. And it is Important to notice that without some further understanding the integral is really ambiguous. According to the assumed law of the secondary wave, the result must actually depend upon the precise radius of the outer boundary of the region of integration, supposed to be exactly circular. This case is, however, at most very special and exceptional. We may usually suppose that a large number of the outer rings are incomplete, so that the integrated term at the upper limit may properly be taken to vanish. If a formal proof be desired, it may be obtained by introducing into the integral a factor such as a hP, in which h is ultimately made to diminish without limit.
When the primary wave is plane, the area of the first Fresnel zone is irXr, and, since the secondary waves vary as the intensity is independent of r, as of course it should be. If, however, the primary wave be spherical, and of radius a at the wavefront of resolution, then we know that at a distance r further on the amplitude of the primary wave will be diminished in the ratio a:(r+a). This may be regarded as a consequence of the altered area of the first Fresnel zone. For, if x be its radius, we have
{(r+§X)2x2}+J {a2x2} tar +a,
so that
x2=Xar/(a+r) nearly.
Since the distance to be travelled by the secondary waves is still r, we see how the effect of the first zone, and therefore of the whole series is proportional to a/(a+r). In like manner may be treated other cases, such as that of a primary wavefront of unequal principal curvatures.
The general explanation of the formation of shadows. may also be conveniently based upon Fresnel's zones. If the point under consideration be so far away from the geometrical shadow that, a large number of the earlier zones are complete,. then the illumination, determined sensibly by the first zone, is the sane as if there were no obstruction at all. If, on the other hand, the point be well immersed in the geometrical shadow, the earlier zones are altogether missing, and, instead of a series of terms beginning with finite numerical magnitude and gradually diminishing to zero, we have now to deal with one of which the terms diminish to zero at both ends. The sum of such a series is very approximately zero, each term being neutralized by the halves of its immediate neighbours, which are of the opposite sign. The question of light or darkness then depends upon whether the series begins or ends abruptly. With few exceptions, abruptness can occur only in the presence of the first term, viz. when the secondary wave of least retardation is unobstructed, or when a ray passes through the point under consideration. According to the undulatory theory the light cannot be regarded strictly as travelling along a ray ; but the existence of an unobstructed ray implies that the system of Fresnel's zones can be commenced, and, if a large number of these zones are fully developed and do not terminate abruptly, the illumination is unaffected by the neighbourhood of obstacles. Intermediate cases in which,a few zones only are formed belong especially to the province of diffraction.
An interesting exception to the general rule that full brightness requires the existence of the first zone occurs when the obstacle assumes the form of a small circular disk parallel to the plane of the incident waves. In the earlier half of the 18th century R. Delisle found, that the centre of the circular shadow was occupied by a bright point of light, but the observation passed into oblivion until S. D. Poisson brought forward as an objection to' Fresnel's theory that it required at the centre of a circular shadow a point as bright as if no obstacle were intervening. If we conceive the primary wave to be broken up at the plane of the disk, a system of Fresnel's zones can be constructed which begin from the circumference; and the first zone external to the disk plays the part ordinarily taken by the centre of the entire system. The whole effect is the
+17r
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 phasereversal without loss of amplitude. R. W. Wood (Phil. Meg., 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 = X(2r)/4(2x),
2X being the diameter of the disk. If 2r =1000 cm., 2x=1 cm., a=6X106 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 •oo1 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(atr,)cos k(atr2) =2 sin kat sin k(rlr2),
rE, r, 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
s 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 strict.z ness 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 ne 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 wavelengths only from the boundary of opaque and transparent parts.
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 convenientlycommence with them on account of their simplicity and' great importance in respect to the theory of optical instruments.
If f 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 wavefront is (§ 2)
dS
gip sink(atp) ,
p being the distance between M and the element dS.
Taking coordinates in the plane of the screen with the centre of the wave as origin, let us represent M by i;, n, and P (where dS is situated) by x, y, z.
Then
pz(x )2+(yn)E+z2, f2 =x2+YE+z! ;
p2 =ff 2x% 2Yn+#z+n2.
In the applications with which we are concerned, t, n are very small quantities; and we may take
p=f )1
f2
yn
At the same time dS may be identified with dxdy, and in the denominator p may be treated as constant and equal to f. Thus the expression for the vibration at M becomes
flfsin k atf+x fYn dxdy . . (1);
and for the intensity, represented by the square of the amplitude,
Iz = a f E [Jf sin kxE f Yndxdy] E E
+a f zf E [ff cos kxE f Y dxdy] . . . . (2).
This expression for the intensity becomes rigorously applicable when f 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 wavefront. 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 objectglass. What is seen through the eyepiece 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. Meg., 1858).
If when the aperture is given, the wavelength (proportional to k1) varies, the composition of the integrals is unaltered, provided and n are taken universely proportional to X. A diminution of X thus leads to a simple proportional shrinkage of the diffraction pattern, attended by an augmentation of brilliancy in proportion to X2.
If the wavelength 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 wavelength 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 la and 1 4a, and for y to be lb, +lb. We readily find (with substitution for Is of 2r/X)
cop sin E f), sin E x
IE= Xz,rEa2e ,,,.Eb2n2
f2X2 f2)2
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 sin Eu/u1, we have to examine the character of this function. It vanishes when u=ma, m being any whole number other than zero. When u =o, it takes the value unity. The maxima occur when
u=tan u, .. (4),
and then .. (5).
sin zu/u2 = cos zu .
To calculate the roots of (5) we may assume u=(m+I)rY=Uy,
so that
where y is a positive quantity which is small when u is large. Substituting this, we find cot y=U—y, whence
y=U (1 { y'U ~.. 'J — 3 — 5 — 315'
This equation is to be solved by successive approximation. It will readily be found that
2 13 146
u=U—y=U—U—1—3U—g 5U_6 1U5U—7—. . (6).
In the first quadrant there is no root after zero, since tan u>u, and in the second quadrant there is none because the signs of u and tan u are opposite. The first root after zero is thus in the third quadrant, corresponding to m =1. Even in this case the series converges sufficiently to give the value of the root with considerable accuracy, while for higher values of m it is all that could be desired. The actual values of u/ar (calculated in another manner by F. M. Schwerd) are 1.4303, 2.4590, 3.4709, 4.4747, 5.4818, 6.4844, &c.
Since the maxima occur when u=(m+2)a nearly, the successive values are not very different from
4 4 4 &c.
GG
The application of these results to (3) shows that the field is brightest at the centre E =o, n =o, viz. at the geometrical image of the radiant point. It is traversed by dark lines whose equations are
E=mfg/a, n=mfx/b.
Within the rectangle formed by pairs of consecutive dark lines, and not far from its centre, the brightness rises to a maximum; but these subsequent maxima are in all cases much inferior to the brightness at the centre of the entire pattern (E=o, n=o).
By the principle of energy the illumination over the entire focal plane must be equal to that over the diffracting area; and thus, in accordance with the suppositions by which (3) was obtained, its value when integrated from E= w to Es= +, and from n = — oo to n= +oo should be equal to ab. This integration, employed originally by P. Kelland (Edin. Trans. 75, p. 315) to determine the absolute intensity of a secondary wave, may be at once effected by means of the known formula
'f'slfliudu = J (sin u
u2 u
du
It will be observed that, while the total intensity is proportional to ab, the intensity at the focal point is proportional to a2b2. If the aperture be increased, not only is the total brightness over the focal plane increased with it, but there is also a concentration of the diffraction pattern. The form of (3) shows immediately that, if a and b be altered, the coordinates of any characteristic point in the pattern vary as a—1 and b—1.
The contraction of the diffraction pattern with increase of aperture is of fundamental importance in connexion with the resolving power of optical instruments. According to common optics, where images are absolute, the diffraction pattern is supposed to be infinitely small, and two radiant points, however near together, form separated images. This is tantamount to an assumption that X is infinitely small. The actual finiteness of X imposes a limit upon the separating or resolving power of an optical instrument.
This indefiniteness of images is sometimes said to be due to diffraction by the edge of the aperture, and proposals have even been made for curing it by causing the transition between the interrupted and transmitted parts of the primary wave to be less abrupt. Such a view of the matter is altogether misleading. What requires explanation is not the imperfection of actual images so much as the possibility of their being as good as we find them.
At the focal point (E=o, 1i=o) all the secondary waves agree in phase, and the intensity is easily expressed, whatever be the form of the aperture. From the general formula (2), if A be the area of aperture,
End of Article: DIFFRACTION OF LIGHT 

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