_[oo]
2[pi] / sin k(at - [rho])x dx
- -------- | ---------------------,
[lambda] _/ 0 [rho]
or, since xdx = [rho]d[rho], k = 2[pi]/[lambda],
_[oo] _ _
/ | |[oo]
-k | sin k(at - [rho])d[rho] = | -cos k(at - [rho])| .
_/r |_ _|r
In order to obtain the effect of the primary wave, as retarded by
traversing the distance r, viz. cos k(at - r), 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
e^-h[rho], in which h is ultimately made to diminish without limit.
When the primary wave is plane, the area of the first Fresnel zone is
[pi][lambda]r, and, since the secondary waves vary as r^-1, 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 wave-front 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 + 1/2[lambda])^2 - x^2} + \/ {a^2 - x^2} = r + a,
so that
x^2 = [lambda]ar/(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 wave-front 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 same 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 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 = [lambda](2r)/4(2x),
2x being the diameter of the disk. If 2r = 1000 cm., 2x = 1 cm.,
[lambda] = 6 X 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 - r1) - cos k(at - r2) = 2 sin kat sin k(r1 - r2),
r2, r1 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.
[Illustration: FIG. 2.]
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.
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 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 wave-front is (S 2)
dS
- ------------- sin k(at - [rho]),
[lambda][rho]
[rho] 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 [xi], [eta], and P (where dS is
situated) by x, y, z. Then
[rho]^2 = (x - [xi])^2 + (y - [eta])^2 + z^2, f^2 = x^2 + y^2 + z^2;
so that
[rho]^2 = f^2 - 2x[xi] - 2y[eta] + [xi]^2 + [eta]^2.
In the applications with which we are concerned, [xi], [eta] are very
small quantities; and we may take
/ x[xi] + y[eta]\
[rho] = f ( 1 - -------------- ).
\ f^2 /
At the same time dS may be identified with dxdy, and in the
denominator [rho] may be treated as constant and equal to f. Thus the
expression for the vibration at M becomes
_ _
1 / / / x[xi] + y[eta]\
- --------------- | | sin k ( at - f + -------------- ) dxdy (1);
[lambda]^2[f]^2 _/_/ \ f /
and for the intensity, represented by the square of the amplitude,
_ _ _ _
1 | / / x[xi] + y[eta] |^2
I^2 = ------------- | | | sin k -------------- dxdy |
[lambda]^2f^2 |_ _/_/ f _|
_ _ _ _
1 | / / x[xi] + y[eta] |^2
+ ------------- | | | cos k -------------- dxdy | (2).
[lambda]^2f^2 |_ _/_/ f _|
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 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 [xi]
and [eta] are taken universely proportional to [lambda]. A diminution
of [lambda] thus leads to a simple proportional shrinkage of the
diffraction pattern, attended by an augmentation of brilliancy in
proportion to [lambda]^-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 2[pi]/[lambda])
[pi]a[xi] [pi]b[eta]
sin^2 --------- sin^2 ----------
a^2b^2 f[lambda] f[lambda]
I^2 = ------------ . ----------------- . ----------------- (3),
f^2[lambda]^2 [pi]^2a^2[xi]^2 [pi]^2b^2[eta]^2
-------------- -------------
f^2[lambda]^2 f^2[lambda]^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^2u/u^2,
we have to examine the character of this function. It vanishes when u
= m[pi], 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
sin^2u/u^2 = cos^2u (5).