Chapter XVIII: Part 18

i.1/2[pi]v^2 / i[pi] i[pi] i[pi] i[pi] i[pi] i[pi] \
C + iS = e ( v - ----- v^3 + ----- ----- v^5 - ----- ----- ----- v^7 + ... ).
\ 3 3 5 3 5 7 /

By separation of real and imaginary parts,

C = M cos 1/2[pi]v^2 - N sin 1/2[pi]v^2 \
S = M sin 1/2[pi]v^2 - N cos 1/2[pi]v^2 / (6)

where

v [pi]^2v^5 [pi]^4v^9
M = - - --------- + --------- - ... (7)
1 3.5 3.5.7.9

[pi]v^3 [pi]^3v^7 [pi]^5v^11
N = ------ - --------- + ------------ ... (8)
1.3 1.3.5.7 1.3.5.7.9.11

These series are convergent for all values of v, but are practically
useful only when v is small.

Expressions suitable for discussion when v is large were obtained by
L. P. Gilbert (_Mem. cour. de l'Acad. de Bruxelles_, 31, p. 1). Taking

1/2[pi]v^2 = u (9),

we may write
_
1 /u e^iu du
C + iS = ------------- | -------- (10).
[sqrt](2[pi]) _/0 [sqrt] u

Again, by a known formula,

_[oo]
1 1 / e^-ux dx
-------- = ---------- | -------- (11).
[sqrt] u [sqrt][pi] _/0 [sqrt]x

Substituting this in (10), and inverting the order of integration, we
get

_[oo] _u
1 / dx / e^u(i - x)
C + iS = ------- | -------- | ----------- dx
[sqrt]2 _/0 [sqrt] x _/0 [sqrt]x

_[oo]
1 / dx e^u(i - x) - 1
= ------- | -------- -------------- dx (12).
[sqrt]2 _/0 [sqrt] x i - x

Thus, if we take

_[oo]
1 / e^-ux [sqrt](x).dx
G = ----------- | ------------------,
[pi][sqrt]2 _/0 1 + x^2

_[oo]
1 / e^-ux dx
H = ----------- | ------------------ (13).
[pi][sqrt]2 _/ [sqrt]x . (1 + x^2)
0

C = 1/2 - G cos u + H sin u, S = 1/2 - G sin u - H cos u (14).

The constant parts in (14), viz. 1/2, may be determined by direct
integration of (12), or from the observation that by their
constitution G and H vanish when u = [oo], coupled with the fact that
C and S then assume the value 1/2.

Comparing the expressions for C, S in terms of M, N, and in terms of
G, H, we find that

G = 1/2 (cos u + sin u) - M, H = 1/2 (cos u - sin u) + N (15),

formulae which may be utilized for the calculation of G, H when u (or
v) is small. For example, when u = 0, M = 0, N = 0, and consequently G
= H = 1/2.

Descending series of the semi-convergent class, available for
numerical calculation when u is moderately large, can be obtained from
(12) by writing x = uy, and expanding the denominator in powers of y.
The integration of the several terms may then be effected by the
formula

_ [oo]
/ -y q-1/2
| e y dy = [Gamma](q + 1/2) = (q - 1/2)(q - 3/2) ... 1/2[sqrt][pi];
_/0

and we get in terms of v

1 1.3.5 1.3.5.9
G = --------- - ---------- + ----------- - (16),
[pi]^2v^3 [pi]^4 v^7 [pi]^6 v^11

1 1.3 1.3.5.7
H = ----- - ---------- + ---------- - (17).
[pi]v [pi]^3 v^5 [pi]^5 v^9

The corresponding values of C and S were originally derived by A. L.
Cauchy, without the use of Gilbert's integrals, by direct integration
by parts.

From the series for G and H just obtained it is easy to verify that

dH dG
-- = - [pi]vG, -- = [pi]vH - 1 (18).
dv dv

We now proceed to consider more particularly the distribution of light
upon a screen PBQ near the shadow of a straight edge A. At a point P
within the geometrical shadow of the obstacle, the half of the wave to
the right of C (fig. 18), the nearest point on the wave-front, is
wholly intercepted, and on the left the integration is to be taken
from s = CA to s = [oo]. If V be the value of v corresponding to CA,
viz.

/ / 2(a + b) \
V= / ( ---------- ).CA, (19),
\/ \ ab[lambda] /

we may write

_[oo] _[oo]
/ / \^2 / / \^2
I^2 = ( | cos 1/2[pi]v^2.dv ) + ( | sin 1/2[pi]v^2.dv ) (20),
\ _/v / \ _/v /

or, according to our previous notation,

I^2 = (1/2 - Cv)^2 + (1/2 - Sv)^2 = G^2 + H^2 (21).

Now in the integrals represented by G and H every element diminishes
as V increases from zero. Hence, as CA increases, viz. as the point P
is more and more deeply immersed in the shadow, the illumination
_continuously_ decreases, and that without limit. It has long been
known from observation that there are no bands on the interior side of
the shadow of the edge.

The law of diminution when V is moderately large is easily expressed
with the aid of the series (16), (17) for G, H. We have ultimately G =
0, H = ([pi]V)^-1, so that

I^2 = 1/[pi]^2V^2,

or the illumination is inversely as the square of the distance from
the shadow of the edge.

For a point Q outside the shadow the integration extends over _more_
than half the primary wave. The intensity may be expressed by

I^2 = (1/2 + Cv)^2 + (1/2 + Sv)^2 (22);

and the maxima and minima occur when

dC dS
(1/2 + C_v) -- + (1/2 + S_v) -- = 0,
dV dV

whence

sin 1/2[pi]V^2 + cos 1/2[pi]V^2 = G (23).

When V = 0, viz. at the edge of the shadow, I^2 = 1/2; when V = [oo],
I^2 = 2, on the scale adopted. The latter is the intensity due to the
uninterrupted wave. The quadrupling of the intensity in passing
outwards from the edge of the shadow is, however, accompanied by
fluctuations giving rise to bright and dark bands. The position of
these bands determined by (23) may be very simply expressed when V is
large, for then sensibly G = 0, and

1/2[pi]V^2 = 3/4[pi] + n[pi] (24),

n being an integer. In terms of [delta], we have from (2)

[delta] = (3/8 + 1/2n)[lambda] (25).

The first maximum in fact occurs when [delta] = 3/8[lambda]
-.0046[lambda], and the first minimum when [delta] = 7/8[lambda]
-.0016[lambda], the corrections being readily obtainable from a table
of G by substitution of the approximate value of V.

The position of Q corresponding to a given value of V, that is, to a
band of given order, is by (19)

a + b / / b[lambda](a + b) \
BQ = ----- AD = V / ( ----------------- ) (26).
a \/ \ 2a /

By means of this expression we may trace the locus of a band of given
order as b varies. With sufficient approximation we may regard BQ and
b as rectangular co-ordinates of Q. Denoting them by x, y, so that AB
is axis of y and a perpendicular through A the axis of x, and
rationalizing (26), we have

2ax^2 - V^2[lambda]y^2 - V^2a[lambda]y = 0,

which represents a hyperbola with vertices at O and A.

From (24), (26) we see that the width of the bands is of the order
[sqrt] {b[lambda](a + b)/a}. From this we may infer the limitation
upon the width of the source of light, in order that the bands may be
properly formed. If [omega] be the apparent magnitude of the source
seen from A, [omega]b should be much smaller than the above quantity,
or

[omega] < [sqrt] {[lambda](a + b)/ab} (27).

If a be very great in relation to b, the condition becomes

[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.

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

In particular, if

2[pi]
f(bt - x) = c sin -------- (bt - x) (5),
[lambda]

we shall have

cdS 2[pi]
[zeta]' = ---------- (1 + cos[theta]) sin[phi]cos -------- (bt - r) (6)."
2[lambda]r [lambda]

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 [phi] as a factor in (6) shows that the relative
intensities of the primary light and of that diffracted in the
direction [theta] 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 [phi] = 1; but, if the primary vibration be in the plane of
diffraction, sin [phi] = cos [theta]. 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 of polarized light are executed in a direction
_perpendicular_ to the plane of polarization.

The factor (1 + cos [theta]) shows in what manner the secondary
disturbance depends upon the direction in which it is propagated with
respect to the front of the primary wave.

If, as suffices for all practical purposes, we limit the application
of the formulae to points in advance of the plane at which the wave is
supposed to be broken up, we may use simpler methods of resolution
than that above considered. It appears indeed that the purely
mathematical question has no definite answer. In illustration of this
the analogous problem for sound may be referred to. Imagine a flexible
lamina to be introduced so as to coincide with the plane at which
resolution is to be effected. The introduction of the lamina (supposed
to be devoid of inertia) will make no difference to the propagation of
plane parallel sonorous waves through the position which it occupies.
At every point the motion of the lamina will be the same as would have
occurred in its absence, the pressure of the waves impinging from
behind being just what is required to generate the waves in front. Now
it is evident that the aerial motion in front of the lamina is
determined by what happens at the lamina without regard to the cause
of the motion there existing. Whether the necessary forces are due to
aerial pressures acting on the rear, or to forces directly impressed
from without, is a matter of indifference. The conception of the
lamina leads immediately to two schemes, according to which a primary
wave may be supposed to be broken up. In the first of these the
element dS, the effect of which is to be estimated, is supposed to
execute its actual motion, while every other element of the plane
lamina is maintained at rest. The resulting aerial motion in front is
readily calculated (see Rayleigh, _Theory of Sound_, S 278); it is
symmetrical with respect to the origin, i.e. independent of [theta].
When the secondary disturbance thus obtained is integrated with
respect to dS over the entire plane of the lamina, the result is
necessarily the same as would have been obtained had the primary wave
been supposed to pass on without resolution, for this is precisely the
motion generated when every element of the lamina vibrates with a
common motion, equal to that attributed to dS. The only assumption
here involved is the evidently legitimate one that, when two systems
of variously distributed motion at the lamina are superposed, the
corresponding motions in front are superposed also.

The method of resolution just described is the simplest, but it is
only one of an indefinite number that might be proposed, and which are
all equally legitimate, so long as the question is regarded as a
merely mathematical one, without reference to the physical properties
of actual screens. If, instead of supposing the _motion_ at dS to be
that of the primary wave, and to be zero elsewhere, we suppose the
_force_ operative over the element dS of the lamina to be that
corresponding to the primary wave, and to vanish elsewhere, we obtain
a secondary wave following quite a different law. In this case the
motion in different directions varies as cos[theta], vanishing at
right angles to the direction of propagation of the primary wave. Here
again, on integration over the entire lamina, the aggregate effect of
the secondary waves is necessarily the same as that of the primary.

In order to apply these ideas to the investigation of the secondary
wave of light, we require the solution of a problem, first treated by
Stokes, viz. the determination of the motion in an infinitely extended
elastic solid due to a locally applied periodic force. If we suppose
that the force impressed upon the element of mass D dx dy dz is

DZ dx dy dz,

being everywhere parallel to the axis of Z, the only change required
in our equations (1), (2) is the addition of the term Z to the second
member of the third equation (2). In the forced vibration, now under
consideration, Z, and the quantities [xi], [eta], [zeta], [delta]
expressing the resulting motion, are to be supposed proportional to
e^int, where i = [sqrt](-1), and n = 2[pi]/[tau], [tau] being the
periodic time. Under these circumstances the double differentiation
with respect to t of any quantity is equivalent to multiplication by
the factor -n^2, and thus our equations take the form

d[delta] \
(b^2[Delta]^2 + n^2)[xi] + (a^2 - b^2) -------- = 0 |
dx |
|
d[delta] |
(b^2[Delta]^2 + n^2)[eta] + (a^2 - b^2) -------- = 0 > (7).
dx |
|
d[delta] |
(b^2[Delta]^2 + n^2)[zeta] + (a^2 - b^2) -------- = -Z |
dx /

It will now be convenient to introduce the quantities.[=omega]1,
[=omega]2, [=omega]3 which express the _rotations_ of the elements of
the medium round axes parallel to those of co-ordinates, in accordance
with the equations

d[xi] d[eta] d[eta] d[zeta]
[=omega]3 = ----- - ------, [=omega]1 = ------ - -------,
dy dx' dz dy

d[zeta] d[xi]
[=omega]2 = ------- - ----- (8).
dx dz

In terms of these we obtain from (7), by differentiation and
subtraction,

(b^2[Delta]^2 + n^2) [=omega]3 = 0 \
(b^2[Delta]^2 + n^2) [=omega]1 = dZ/dy > (9).
(b^2[Delta]^2 + n^2) [=omega]2 = -dZ/dx /

The first of equations (9) gives

[=omega]3 = 0 (10).

For =[omega]1, we have
_ _ _ -ikr
1 / / / dZ e
[=omega]1 = -------- | | | -- ----- dx dy dz (11),
4[pi]b^2 _/_/_/ dy r

where r is the distance between the element dx dy dz and the point
where [=omega]1 is estimated, and

k = n/b = 2[pi]/[lambda] (12),

[lambda] being the wave-length.

(This solution may be verified in the same manner as Poisson's
theorem, in which k = 0.)

We will now introduce the supposition that the force Z acts only
within a small space of volume T, situated at (x, y, z), and for
simplicity suppose that it is at the origin of co-ordinates that the
rotations are to be estimated. Integrating by parts in (11), we get

_ -ikr _ _ _
/ e dZ | Ze^-ikr | / d / e^-ikr\
| ------ -- dy = | ------- | - | Z -- ( ------- ) dy,
_/ r dy |_ r _| _/ dy \ r /

in which the integrated terms at the limits vanish, Z being finite
only within the region T. Thus

_ _ _ -ikr
1 / / / d /e^ \
[=omega]1 = ------- | | | Z -- ( -------- ) dx dy dz.
4[pi]b^2 _/_/_/ dy \ r /

Since the dimensions of T are supposed to be very small in comparison
with [lambda], the factor d/dy (e^-ikr / r) is sensibly constant; so
that, if Z stand for the mean value of Z over the volume T, we may
write

TZ y d / e^-ikr \
[=omega]1 = -------- . - . -- ( ------ ) (13).
4[pi]b^2 r dr \ r /

In like manner we find

TZ x d / e^-ikr \
[=omega]2 = -------- . - . -- ( ------- ) (14).
4[pi]b^2 r dr \ r /

From (10), (13), (14) we see that, as might have been expected, the
rotation at any point is about an axis perpendicular both to the
direction of the force and to the line joining the point to the source
of disturbance. If the resultant rotation be [omega], we have

TZ [sqrt](x^2 + y^2) d /e^-ikr\
[=omega] = ------- . ----------------- . -- ( ------ ) =
4[pi]b^2 r dr \ r /

TZ sin[phi] d /e^-ikr\
= ----------- -- ( ------ ),
4[pi]b^2 dr \ r /

[phi] denoting the angle between r and z. In differentiating
e^(-ikr)/r with respect to r, we may neglect the term divided by r^2 as
altogether insensible, kr being an exceedingly great quantity at any
moderate distance from the origin of disturbance. Thus

-ik.TZ sin[phi] /e^-ikr\
[=omega] = --------------- . ( ------ ) (15),
4[pi]b^2 \ r /

which completely determines the rotation at any point. For a
disturbing force of given integral magnitude it is seen to be
everywhere about an axis perpendicular to r and the direction of the
force, and in magnitude dependent only upon the angle ([phi]) between
these two directions and upon the distance (r).

The intensity of light is, however, more usually expressed in terms of
the actual displacement in the plane of the wave. This displacement,
which we may denote by [zeta]', is in the plane containing z and r,
and perpendicular to the latter. Its connexion with [=omega]is
expressed by [=omega] = d[zeta]'/dr; so that

TZ sin [phi] /e^-ikr\
[zeta]' = ----------- . ( ------ ) (16),
4[pi]b^2 \ r /

where the factor e^int is restored.

Retaining only the real part of (16), we find, as the result of a
local application of force equal to

DTZ cos nt (17),

the disturbance expressed by

TZ sin [phi] /cos(nt - kr)\
[zeta]' = ------------ . ( ------------ ) (18).
4[pi]b^2 \ r /

The occurrence of sin [phi] shows that there is no disturbance
radiated in the direction of the force, a feature which might have
been anticipated from considerations of symmetry.

We will now apply (18) to the investigation of a law of secondary
disturbance, when a primary wave

[zeta] = sin(nt - kx) (19)

is supposed to be broken up in passing the plane x = 0. The first step
is to calculate the force which represents the reaction between the
parts of the medium separated by x = 0. The force operative upon the
positive half is parallel to OZ, and of amount per unit of area equal
to

-b^2D d[zeta]/dx = b^2kD cos nt;

and to this force acting over the whole of the plane the actual motion
on the positive side may be conceived to be due. The secondary
disturbance corresponding to the element dS of the plane may be
supposed to be that caused by a force of the above magnitude acting
over dS and vanishing elsewhere; and it only remains to examine what
the result of such a force would be.

Now it is evident that the force in question, supposed to act upon the
positive half only of the medium, produces just double of the effect
that would be caused by the same force if the medium were undivided,
and on the latter supposition (being also localized at a point) it
comes under the head already considered. According to (18), the effect
of the force acting at dS parallel to OZ, and of amount equal to

2b^2kD dS cos nt,

will be a disturbance

dS sin [phi]
[zeta]' = ------------ cos(nt - kr) (20),
[lambda]r

regard being had to (12). This therefore expresses the secondary
disturbance at a distance r and in a direction making an angle [phi]
with OZ (the direction of primary vibration) due to the element dS of
the wave-front.

The proportionality of the secondary disturbance to sin [phi] is
common to the present law and to that given by Stokes, but here there
is no dependence upon the angle [theta] between the primary and
secondary rays. The occurrence of the factor [lambda]r^-1, and the
necessity of supposing the phase of the secondary wave accelerated by
a quarter of an undulation, were first established by Archibald Smith,
as the result of a comparison between the primary wave, supposed to
pass on without resolution, and the integrated effect of all the
secondary waves (S 2). The occurrence of factors such as sin [phi], or
1/2(1 + cos [theta]), in the expression of the secondary wave has no
influence upon the result of the integration, the effects of all the
elements for which the factors differ appreciably from unity being
destroyed by mutual interference.

The choice between various methods of resolution, all mathematically
admissible, would be guided by physical considerations respecting the
mode of action of obstacles. Thus, to refer again to the acoustical
analogue in which plane waves are incident upon a perforated rigid
screen, the circumstances of the case are best represented by the
first method of resolution, leading to symmetrical secondary waves, in
which the normal motion is supposed to be zero over the unperforated
parts. Indeed, if the aperture is very small, this method gives the
correct result, save as to a constant factor. In like manner our
present law (20) would apply to the kind of obstruction that would be
caused by an actual physical division of the elastic medium, extending
over the whole of the area supposed to be occupied by the intercepting
screen, but of course not extending to the parts supposed to be
perforated.

On the electromagnetic theory, the problem of diffraction becomes
definite when the properties of the obstacle are laid down. The
simplest supposition is that the material composing the obstacle is
perfectly conducting, i.e. perfectly reflecting. On this basis A. J.
W. Sommerfeld (_Math. Ann._, 1895, 47, p. 317), with great
mathematical skill, has solved the problem of the shadow thrown by a
semi-infinite plane screen. A simplified exposition has been given by
Horace Lamb (_Proc. Lond. Math. Soc._, 1906, 4, p. 190). It appears
that Fresnel's results, although based on an imperfect theory, require
only insignificant corrections. Problems not limited to two
dimensions, such for example as the shadow of a circular disk, present
great difficulties, and have not hitherto been treated by a rigorous
method; but there is no reason to suppose that Fresnel's results would
be departed from materially. (R.)

FOOTNOTES:

[1] The descending series for J0(z) appears to have been first given
by Sir W. Hamilton in a memoir on "Fluctuating Functions," _Roy.
Irish Trans._, 1840.

[2] Airy, loc. cit. "Thus the magnitude of the central spot is
diminished, and the brightness of the rings increased, by covering
the central parts of the object-glass."

[3] _"Man kann daraus schliessen, was moglicher Weise durch
Mikroskope noch zu sehen ist. Ein mikroskopischer Gegenstand z. B,
dessen Durchmesser = ([lambda]) ist, und der aus zwei Theilen
besteht, kann nicht mehr als aus zwei Theilen bestehend erkannt
werden. Dieses zeigt uns eine Grenze des Sehvermogens durch
Mikroskope"_ (_Gilbert's Ann._ 74, 337). Lord Rayleigh has recorded
that he was himself convinced by Fraunhofer's reasoning at a date
antecedent to the writings of Helmholtz and Abbe.

[4] The last sentence is repeated from the writer's article "Wave
Theory" in the 9th edition of this work, but A. A. Michelson's
ingenious echelon grating constitutes a realization in an unexpected
manner of what was thought to be impracticable.--[R.]

[5] Compare also F. F. Lippich, _Pogg. Ann._ cxxxix. p. 465, 1870;
Rayleigh, _Nature_ (October 2, 1873).

[6] The power of a grating to construct light of nearly definite
wave-length is well illustrated by Young's comparison with the
production of a musical note by reflection of a sudden sound from a
row of palings. The objection raised by Herschel (_Light_, S 703) to
this comparison depends on a misconception.

[7] It must not be supposed that errors of this order of magnitude
are unobjectionable in all cases. The position of the middle of the
bright band representative of a mathematical line can be fixed with a
spider-line micrometer within a small fraction of the width of the
band, just as the accuracy of astronomical observations far
transcends the separating power of the instrument.

[8] "In the same way we may conclude that in flat gratings any
departure from a straight line has the effect of causing the dust in
the slit and the spectrum to have different foci--a fact sometimes
observed." (Rowland, "On Concave Gratings for Optical Purposes,"
_Phil. Mag._, September 1883).

[9] On account of inequalities in the atmosphere giving a variable
refraction, the light from a star would be irregularly distributed
over a screen. The experiment is easily made on a laboratory scale,
with a small source of light, the rays from which, in their course
towards a rather distant screen, are disturbed by the neighbourhood
of a heated body. At a moment when the eye, or object-glass of a
telescope, occupies a dark position, the star vanishes. A fraction of
a second later the aperture occupies a bright place, and the star
reappears. According to this view the chromatic effects depend
entirely upon atmospheric dispersion.

[10] In experiment a line of light is sometimes substituted for a
point in order to increase the illumination. The various parts of the
line are here _independent_ sources, and should be treated
accordingly. To assume a cylindrical form of primary wave would be
justifiable only when there is synchronism among the secondary waves
issuing from the various centres.

[11] 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.

DIFFUSION (from the Lat. _diffundere; dis-_, asunder, and _fundere_, to pour out), in general, a spreading out, scattering or circulation; in physics the term is applied to a special phenomenon, treated below.

1. _General Description._--When two different substances are placed in contact with each other they sometimes remain separate, but in many cases a gradual mixing takes place. In the case where both the substances are gases the process of mixing continues until the result is a uniform mixture. In other cases the proportions in which two different substances can mix lie between certain fixed limits, but the mixture is distinguished from a chemical compound by the fact that between these limits the composition of the mixture is capable of continuous variation, while in chemical compounds, the proportions of the different constituents can only have a discrete series of numerical values, each different ratio representing a different compound. If we take, for example, air and water in the presence of each other, air will become dissolved in the water, and water will evaporate into the air, and the proportions of either constituent absorbed by the other will vary continuously. But a limit will come when the air will absorb no more water, and the water will absorb no more air, and throughout the change a definite surface of separation will exist between the liquid and the gaseous parts. When no surface of separation ever exists between two substances they must necessarily be capable of mixing in all proportions. If they are not capable of mixing in all proportions a discontinuous change must occur somewhere between the regions where the substances are still unmixed, thus giving rise to a surface of separation.

The phenomena of mixing thus involves the following processes:--(1) A motion of the substances relative to one another throughout a definite _region_ of space in which mixing is taking place. This relative motion is called "diffusion." (2) The passage of portions of the mixing substances across the _surface_ of separation when such a surface exists. These surface actions are described under various terms such as solution, evaporation, condensation and so forth. For example, when a soluble salt is placed in a liquid, the process which occurs at the surface of the salt is called "solution," but the salt which enters the liquid by solution is transported from the surface into the interior of the liquid by "diffusion."

Diffusion may take place in solids, that is, in regions occupied by matter which continues to exhibit the properties of the solid state. Thus if two liquids which can mix are separated by a membrane or partition, the mixing may take place through the membrane. If a solution of salt is separated from pure water by a sheet of parchment, part of the salt will pass through the parchment into the water. If water and glycerin are separated in this way most of the water will pass into the glycerin and a little glycerin will pass through in the opposite direction, a property frequently used by microscopists for the purpose of gradually transferring minute algae from water into glycerin. A still more interesting series of examples is afforded by the passage of gases through partitions of metal, notably the passage of hydrogen through platinum and palladium at high temperatures. When the process is considered with reference to a membrane or partition taken as a whole, the passage of a substance from one side to the other is commonly known as "osmosis" or "transpiration" (see SOLUTION), but what occurs in the material of the membrane itself is correctly described as diffusion.

Simple cases of diffusion are easily observed qualitatively. If a solution of a coloured salt is carefully introduced by a funnel into the bottom of a jar containing water, the two portions will at first be fairly well defined, but if the mixture can exist in all proportions, the surface of separation will gradually disappear; and the rise of the colour into the upper part and its gradual weakening in the lower part, may be watched for days, weeks or even longer intervals. The diffusion of a strong aniline colouring matter into the interior of gelatine is easily observed, and is commonly seen in copying apparatus. Diffusion of gases may be shown to exist by taking glass jars containing vapours of hydrochloric acid and ammonia, and placing them in communication with the heavier gas downmost. The precipitation of ammonium chloride shows that diffusion exists, though the chemical action prevents this example from forming a typical case of diffusion. Again, when a film of Canada balsam is enclosed between glass plates, the disappearance during a few weeks of small air bubbles enclosed in the balsam can be watched under the microscope.

In fluid media, whether liquids or gases, the process of mixing is greatly accelerated by stirring or agitating the fluids, and liquids which might take years to mix if left to themselves can thus be mixed in a few seconds. It is necessary to carefully distinguish the effects of agitation from those of diffusion proper. By shaking up two liquids which do not mix we split them up into a large number of different portions, and so greatly increase the area of the surface of separation, besides decreasing the thicknesses of the various portions. But even when we produce the appearance of a uniform turbid mixture, the small portions remain quite distinct. If however the fluids can really mix, the final process must in every case depend on diffusion, and all we do by shaking is to increase the sectional area, and decrease the thickness of the diffusing portions, thus rendering the completion of the operation more rapid. If a gas is shaken up in a liquid the process of absorption of the bubbles is also accelerated by capillary action, as occurs in an ordinary sparklet bottle. To state the matter precisely, however finely two fluids have been subdivided by agitation, the molecular constitution of the different portions remains unchanged. The ultimate process by which the individual molecules of two different substances become mixed, producing finally a homogeneous mixture, is in every case diffusion. In other words, diffusion is that relative motion of the molecules of two different substances by which the proportions of the molecules in any region containing a finite number of molecules are changed.

In order, therefore, to make accurate observations of diffusion in
fluids it is necessary to guard against any cause which may set up
currents; and in some cases this is exceedingly difficult. Thus, if
gas is absorbed at the upper surface of a liquid, and if the gaseous
solution is heavier than the pure liquid, currents may be set up, and
a steady state of diffusion may cease to exist. This has been tested
experimentally by C. G. von Hufner and W. E. Adney. The same thing may
happen when a gas is evolved into a liquid at the surface of a solid
even if no bubbles are formed; thus if pieces of aluminium are placed
in caustic soda, the currents set up by the evolution of hydrogen are
sufficient to set the aluminium pieces in motion, and it is probable
that the motions of the Diatomaceae are similarly caused by the
evolution of oxygen. In some pairs of substances diffusion may take
place more rapidly than in others. Of course the progress of events in
any experiment necessarily depends on various causes, such as the size
of the containing vessels, but it is easy to see that when experiments
with different substances are carried out under similar conditions,
however these "similar conditions" be defined, the rates of diffusion
must be capable of numerical comparison, and the results must be
expressible in terms of at least one physical quantity, which for any
two substances can be called their coefficient of diffusion. How to
select this quantity we shall see later.

2 _Quantitative Methods of observing Diffusion._--The simplest plan of determining the progress of diffusion between two liquids would be to draw off and examine portions from different strata at some stage in the process; the disturbance produced would, however, interfere with the subsequent process of diffusion, and the observations could not be continued. By placing in the liquid column hollow glass beads of different average densities, and observing at what height they remain suspended, it is possible to trace the variations of density of the liquid column at different depths, and different times. In this method, which was originally introduced by Lord Kelvin, difficulties were caused by the adherence of small air bubbles to the beads.

In general, optical methods are the most capable of giving exact results, and the following may be distinguished, (a) _By refraction in a horizontal plane._ If the containing vessel is in the form of a prism, the deviation of a horizontal ray of light in passing through the prism determines the index of refraction, and consequently the density of the stratum through which the ray passes, (b) _By refraction in a vertical plane._ Owing to the density varying with the depth, a horizontal ray entering the liquid also undergoes a small vertical deviation, being bent downwards towards the layers of greater density. The observation of this vertical deviation determines not the actual density, but its rate of variation with the depth, i.e. the "density gradient" at any point, (c) _By the saccharimeter._ In the cases of solutions of sugar, which cause rotation of the plane of polarized light, the density of the sugar at any depth may be determined by observing the corresponding angle of rotation, this was done originally by W. Voigt.

3. _Elementary Definitions of Coefficient of Diffusion._--The simplest case of diffusion is that of a substance, say a gas, diffusing in the interior of a homogeneous solid medium, which remains at rest, when no external forces act on the system. We may regard it as the result of experience that: (1) if the density of the diffusing substance is everywhere the same no diffusion takes place, and (2) if the density of the diffusing substance is different at different points, diffusion will take place from places of greater to those of lesser density, and will not cease until the density is everywhere the same. It follows that the rate of flow of the diffusing substance at any point in any direction must depend on the density gradient at that point in that direction, i.e. on the rate at which the density of the diffusing substance decreases as we move in that direction. We may define the _coefficient of diffusion_ as the ratio of the total mass per unit area which flows across any small section, to the rate of decrease of the density per unit distance in a direction perpendicular to that section.

In the case of steady diffusion parallel to the axis of x, if [rho] be
the density of the diffusing substance, and q the mass which flows
across a unit of area in a plane perpendicular to the axis of x, then
the density gradient is -d[rho]/dx and the ratio of q to this is
called the "coefficient of diffusion." By what has been said this
ratio remains finite, however small the actual gradient and flow may
be., and it is natural to assume, at any rate as a first
approximation, that it is constant as far as the quantities in
question are concerned. Thus if the coefficient of diffusion be
denoted by K we have q= -K(d[rho]/dx).

Further, the rate at which the quantity of substance is increasing in
an element between the distances x and x+dx is equal to the difference
of the rates of flow in and out of the two faces, whence as in
hydrodynamics, we have d[rho]/dt =-dq/dx.

It follows that the equation of diffusion in this case assumes the
form

d[rho] d / d[rho] \
------ = -- ( K ------ ),
dt dx \ dx /

which is identical with the equations representing conduction of heat,
flow of electricity and other physical phenomena. For motion in three
dimensions we have in like manner

d[rho] d / d[rho]\ d / d[rho]\ d / d[rho]\
------ = -- ( K ------ ) + -- ( K ------ ) + -- ( K ------ );
dt dx \ dx / dy \ dy / dz \ dz /

and the corresponding equations in electricity and heat for
anisotropic substances would be available to account for any parallel
phenomena, which may arise, or might be conceived, to exist in
connexion with diffusion through a crystalline solid.

In the case of a very dilute solution, the coefficient of diffusion of the dissolved substance can be defined in the same way as when the diffusion takes place in a solid, because the effects of diffusion will not have any perceptible influence on the solvent, and the latter may therefore be regarded as remaining practically at rest. But in most cases of diffusion between two fluids, both of the fluids are in motion, and hence there is far greater difficulty in determining the motion, and even in defining the coefficient of diffusion. It is important to notice in the first instance, that it is only the relative motion of the two substances which constitutes diffusion. Thus when a current of air is blowing, under ordinary circumstances the changes which take place are purely mechanical, and do not depend on the separate diffusions of the oxygen and nitrogen of which the air is mainly composed. It is only when two gases are flowing with unequal velocity, that is, when they have a relative motion, that these changes of relative distribution, which are called diffusion, take place. The best way out of the difficulty is to investigate the separate motions of the two fluids, taking account of the mechanical actions exerted on them, and supposing that the mutual action of the fluids causes either fluid to resist the relative motion of the other.

4. _The Coefficient of Resistance._--Let us call the two diffusing fluids A and B. If B were absent, the motion of the fluid A would be determined entirely by the variations of pressure of the fluid A, and by the external forces, such as that due to gravity acting on A. Similarly if A were absent, the motion of B would be determined entirely by the variations of pressure due to the fluid B, and by the external forces acting on B. When both fluids are mixed together, each fluid tends to resist the relative motion of the other, and by the law of equality of action and reaction, the resistance which A experiences from B is everywhere equal and opposite to the resistance which B experiences from A. If the amount of this resistance per unit volume be divided by the relative velocity of the two fluids, and also by the product of their densities, the quotient is called the "coefficient of resistance." If then [rho]1, [rho]2 are the densities cf the two fluids, u1, u2 their velocities, C the coefficient of resistance, then the portion of the fluid A contained in a small element of volume v will experience from the fluid B a resistance C[rho]1[rho]2v(u1- u2), and the fluid B contained in the same volume element will experience from the fluid A an equal and opposite resistance, C[rho]1[rho]2v(u2 - u1).

This definition implies the following laws of resistance to diffusion, which must be regarded as based on experience, and not as self-evident truths: (1) each fluid tends to assume, so far as diffusion is concerned, the same equuibrium distribution that it would assume if its motion were unresisted by the presence of the other fluid. (Of course, the mutual attraction of gravitation of the two fluids might affect the final distribution, but this is practically negligible. Leaving such actions as this out of account the following statement is correct.) In a state of equilibrium, the density of each fluid at any point thus depends only on the partial pressure of that fluid alone, and is the same as if the other fluids were absent. It does not depend on the partial pressures of the other fluids. If this were not the case, the resistance to diffusion would be analogous to friction, and would contain terms which were independent of the relative velocity u2 - u1. (2) For slow motions the resistance to diffusion is (approximately at any rate) proportional to the relative velocity. (3) The coefficient of resistance C is not necessarily always constant; it may, for example, and, in general, does, depend on the temperature.

If we form the equations of hydrodynamics for the different fluids
occurring in any mixture, taking account of diffusion, but neglecting
viscosity, and using suffixes 1, 2 to denote the separate fluids,
these assume the form given by James Clerk Maxwell ("Diffusion," in
_Ency. Brit._, 9th ed.):--

Du1 dp1
[rho] --- + --- - X1[rho]1 + C12[rho]1[rho]2(u1 - u2) + &c. = 0,
Dt dx

where

Du1 du1 du1 du1 du1
--- = --- + u1 --- + v1 --- + w1 ---,
Dt dt dx dy dz

and these equations imply that when diffusion and other motions cease,
the fluids satisfy the separate conditions of equilibrium dp1/dx -
X1[rho]1 = 0. The assumption made in the following account is that
terms such as Du1/Dt may be neglected in the cases considered.