[omega] < [sqrt] ([lambda]/b) (28).
so that if b is to be moderately great (1 metre), the apparent
magnitude of the sun must be greatly reduced before it can be used as
a source. The values of V for the maxima and minima of intensity, and
the magnitudes of the latter, were calculated by Fresnel. An extract
from his results is given in the accompanying table.
+--------------------+----------+------------+
| | V | I^2 |
+--------------------+----------+------------+
| First maximum | 1.2172 | 2.7413 |
| First minimum | 1.8726 | 1.5570 |
| Second maximum | 2.3449 | 2.3990 |
| Second minimum | 2.7392 | 1.6867 |
| Third maximum. | 3.0820 | 2.3022 |
| Third minimum | 3.3913 | 1.7440 |
+--------------------+----------+------------+
A very thorough investigation of this and other related questions,
accompanied by fully worked-out tables of the functions concerned,
will be found in a paper by E. Lommel (_Abh. bayer. Akad. d. Wiss._
II. CI., 15, Bd., iii. Abth., 1886).
When the functions C and S have once been calculated, the discussion
of various diffraction problems is much facilitated by the idea, due
to M. A. Cornu (_Journ. de Phys._, 1874, 3, p. 1; a similar suggestion
was made independently by G. F. Fitzgerald), of exhibiting as a curve
the relationship between C and S, considered as the rectangular
co-ordinates (x, y) of a point. Such a curve is shown in fig. 19,
where, according to the definition (5) of C, S,
_ v _ v
/ /
x = | cos 1/2[pi]v^2.dv, y = | sin 1/2[pi]v^2.dv (29).
_/0 _/0
The origin of co-ordinates O corresponds to v = 0; and the asymptotic
points J, J', round which the curve revolves in an ever-closing
spiral, correspond to v = [+-][oo].
The intrinsic equation, expressing the relation between the arc
[sigma] (measured from O) and the inclination [phi] of the tangent at
any points to the axis of x, assumes a very simple form. For
dx = cos 1/2[pi]v^2.dv, dy = sin 1/2[pi]v^2.dv;
so that
_
/
[sigma] = | [sqrt] (dx^2 + dy^2) = v, (30),
_/
[phi] = tan^-1 (dy/dx) = 1/2[pi]v^2 (31).
Accordingly,
[phi] = 1/2[pi][sigma]^2 (32);
and for the curvature,
d[phi]/d[sigma] = [pi][sigma] (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.
[Illustration: 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)
d[sigma] = 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 2[pi][delta]/[lambda], or [phi]. 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 ([sigma]2 - [sigma]1).
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 oi J, J' being (1/2, 1/2),
(-1/2, -1/2), I^2 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."[11]
II. _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 [chi], [eta], [zeta] 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
d^2[xi] /d^2[xi] d^2[xi] d^2[xi]\ d^2 /d^2[xi] d^2[eta] d^2[zeta]\
------ = b^2( ------ + -------- + ------- ) + (a^2 - b^2)---( ------- + -------- + --------- ),
dt^2 \ dx^2 dy^2 dz^2 / dx \ dx^2 dy^2 dz^2 /
where a2 and b2 denote the two arbitrary constants. Put for shortness
d^2[xi] d^2[eta] d^2[zeta]
------- + -------- + --------- = [delta] (1),
dx^2 dy^2 dz^2
and represent by [Delta]^2[chi] the quantity multiplied by b^2.
According to this notation, the three equations of motion are
d^2[xi] d[delta] \
------- = b^2[Delta]^2[xi] + (a^2 - b^2) -------- |
dt^2 dx |
|
d^2[eta] d[delta] |
-------- = b^2[Delta]^2[eta] + (a^2 - b^2) -------- > (2).
dt^2 dy |
|
d^2[zeta] d[delta] |
--------- = b^2[Delta]^2[zeta] + (a^2 - b^2) -------- |
dt^2 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 [delta] vanishes; but,
in order that equations (2) may preserve their generality, we must
suppose a at the same time to become infinite, and replace a^2[delta]
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 [xi] = 0, [eta] = 0, while [zeta] 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
d^2[zeta] d^2[zeta]
--------- = --------- (3),
dt^2 dx^2
of which the solution is
[zeta] = f(bt - x) (4),
where f is an arbitrary function.
The question as to the law of the secondary waves is thus answered by
Stokes. "Let [xi] = 0, [eta] = 0, [zeta] = f(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 [theta] with the direction of
propagation of the incident light, or the axis of x, and [phi] 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 [zeta]' be the displacement at O, reckoned
positive in the direction nearest to that in which the incident
vibrations are reckoned positive,
dS
[zeta]' = ------ ( 1 + cos[theta]) sin[phi] f'(bt - r).
4[pi]r