u = (m + 1/2)[pi] - y = U - y,
where y is a positive quantity which is small when u is large.
Substituting this, we find cot y = U - y, whence
1 / y y- \ y^3 2y^5 17y^7
y = - ( 1 + - + --- + ... ) - --- ---- - -----.
U \ U U^2 / 3 15 315
This equation is to be solved by successive approximation. It will
readily be found that
2 13 146
u = U - y = U - U^-1 - -- U^-3 - -- U^-5 - --- U^-7 - ... (6).
3 15 105
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/[pi] (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 + 1/2)[pi] nearly, the successive
values are not very different from
4 4 4
-------, ------, --------, &c.
9[pi]^2 25[pi] 49[pi]^2
The application of these results to (3) shows that the field is
brightest at the centre [xi] = 0, [eta] = 0, viz. at the geometrical
image of the radiant point. It is traversed by dark lines whose
equations are
[xi] = mf[lambda]/a, [eta] = mf[lambda]/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 ([xi] = 0, [eta] = 0).
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 [xi] = [oo] to [xi] = +[oo], and from [eta] =
-[oo] to [eta] = +[oo] should be equal to ab. This integration,
employed originally by P. Kelland (_Edin. Trans._ 15, p. 315) to
determine the absolute intensity of a secondary wave, may be at once
effected by means of the known formula
_+[oo] _+[oo]
/ sin^2u / sin u
| ----- du = | ----- du = [pi].
_/ u^2 _/ u
-[oo] -[oo]
It will be observed that, while the total intensity is proportional to
ab, the intensity at the focal point is proportional to a^2b^2. 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 co-ordinates 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 [lambda] is infinitely small.
The actual finiteness of [lambda] 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 ([xi] = 0, [eta] = 0) 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,
I0^2 = A^2/[lambda]^2f^2 (7).
The formation of a sharp image of the radiant point requires that the
illumination become insignificant when [xi], [eta] attain small
values, and this insignificance can only arise as a consequence of
discrepancies of phase among the secondary waves from various parts of
the aperture. So long as there is no sensible discrepancy of phase
there can be no sensible diminution of brightness as compared with
that to be found at the focal point itself. We may go further, and lay
it down that there can be no considerable loss of brightness until the
difference of phase of the waves proceeding from the nearest and
farthest parts of the aperture amounts to 1/4[lambda].
When the difference of phase amounts to [lambda], we may expect the
resultant illumination to be very much reduced. In the particular case
of a rectangular aperture the course of things can be readily
followed, especially if we conceive f to be infinite. In the direction
(suppose horizontal) for which [eta] = 0, [xi]/f = sin [theta], the
phases of the secondary waves range over a complete period when sin
[theta] = [lambda]/a, and, since all parts of the horizontal aperture
are equally effective, there is in this direction a complete
compensation and consequent absence of illumination. When sin [theta]
= 3/2[lambda]/a, the phases range one and a half periods, and there
is revival of illumination. We may compare the brightness with that in
the direction [theta] = 0. The phase of the resultant amplitude is the
same as that due to the central secondary wave, and the discrepancies
of phase among the components reduce the amplitude in the proportion
_+3/2[pi]
1 /
----- | cos [phi] d[phi]: 1,
3[pi] _/-3/2[pi]
or -2/3[pi]:1; so that the brightness in this direction is 4/9[pi]^2 of
the maximum at [theta] = 0. In like manner we may find the
illumination in any other direction, and it is obvious that it
vanishes when sin [theta] is any multiple of [lamba]/a.
The reason of the augmentation of resolving power with aperture will
now be evident. The larger the aperture the smaller are the angles
through which it is necessary to deviate from the principal direction
in order to bring in specified discrepancies of phase--the more
concentrated is the image.
In many cases the subject of examination is a luminous line of uniform
intensity, the various points of which are to be treated as
independent sources of light. If the image of the line be [xi] = 0,
the intensity at any point [xi], [eta] of the diffraction pattern may
be represented by
[pi]a[xi]
_+[oo] sin^2---------
/ a^2b [lambda]f
| I^2d[eta] = --------- ------------- (8),
_/ [lambda]f [pi]^2a^2[xi]^2
-[oo] ---------------
[lambda]^2f^2
the same law as obtains for a luminous point when horizontal
directions are alone considered. The definition of a fine vertical
line, and consequently the resolving power for contiguous vertical
lines, is thus _independent of the vertical aperture of the
instrument_, a law of great importance in the theory of the
spectroscope.
The distribution of illumination in the image of a luminous line is
shown by the curve ABC (fig. 3), representing the value of the
function sin^2u/u^2 from u = 0 to u = 2[pi]. The part corresponding to
negative values of u is similar, OA being a line of symmetry.
[Illustration: Fig. 3.]
Let us now consider the distribution of brightness in the image of a
double line whose components are of equal strength, and at such an
angular interval that the central line in the image of one coincides
with the first zero of brightness in the image of the other. In fig. 3
the curve of brightness for one component is ABC, and for the other
OA'C'; and the curve representing half the combined brightnesses is
E'BE. The brightness (corresponding to B) midway between the two
central points AA' is .8106 of the brightness at the central points
themselves. We may consider this to be about the limit of closeness at
which there could be any decided appearance of resolution, though
doubtless an observer accustomed to his instrument would recognize the
duplicity with certainty. The obliquity, corresponding to u = [pi], is
such that the phases of the secondary waves range over a complete
period, i.e. such that the projection of the horizontal aperture upon
this direction is one wave-length. We conclude that a _double line
cannot be fairly resolved unless its components subtend an angle
exceeding that subtended by the wave-length of light at a distance
equal to the horizontal aperture_. This rule is convenient on account
of its simplicity; and it is sufficiently accurate in view of the
necessary uncertainty as to what exactly is meant by resolution.
If the angular interval between the components of a double line be
half as great again as that supposed in the figure, the brightness
midway between is .1802 as against 1.0450 at the central lines of each
image. Such a falling off in the middle must be more than sufficient
for resolution. If the angle subtended by the components of a double
line be twice that subtended by the wave-length at a distance equal to
the horizontal aperture, the central bands are just clear of one
another, and there is a line of absolute blackness in the middle of
the combined images.
The resolving power of a telescope with circular or rectangular
aperture is easily investigated experimentally. The best object for
examination is a grating of fine wires, about fifty to the inch,
backed by a sodium flame. The object-glass is provided with diaphragms
pierced with round holes or slits. One of these, of width equal, say,
to one-tenth of an inch, is inserted in front of the object-glass, and
the telescope, carefully focused all the while, is drawn gradually
back from the grating until the lines are no longer seen. From a
measurement of the maximum distance the least angle between
consecutive lines consistent with resolution may be deduced, and a
comparison made with the rule stated above.
Merely to show the dependence of resolving power on aperture it is not
necessary to use a telescope at all. It is sufficient to look at wire
gauze backed by the sky or by a flame, through a piece of blackened
cardboard, pierced by a needle and held close to the eye. By varying
the distance the point is easily found at which resolution ceases; and
the observation is as sharp as with a telescope. The function of the
telescope is in fact to allow the use of a wider, and therefore more
easily measurable, aperture. An interesting modification of the
experiment may be made by using light of various wave-lengths.
Since the limitation of the width of the central band in the image of
a luminous line depends upon discrepancies of phase among the
secondary waves, and since the discrepancy is greatest for the waves
which come from the edges of the aperture, the question arises how far
the operation of the central parts of the aperture is advantageous. If
we imagine the aperture reduced to two equal narrow slits bordering
its edges, compensation will evidently be complete when the projection
on an oblique direction is equal to 1/2[lambda], instead of [lambda] as
for the complete aperture. By this procedure the width of the central
band in the diffraction pattern is halved, and so far an advantage is
attained. But, as will be evident, the bright bands bordering the
central band are now not inferior to it in brightness; in fact, a band
similar to the central band is reproduced an indefinite number of
times, so long as there is no sensible discrepancy of phase in the
secondary waves proceeding from the various parts of the _same_ slit.
Under these circumstances the narrowing of the band is paid for at a
ruinous price, and the arrangement must be condemned altogether.
A more moderate suppression of the central parts is, however,
sometimes advantageous. Theory and experiment alike prove that a
double line, of which the components are equally strong, is better
resolved when, for example, one-sixth of the horizontal aperture is
blocked off by a central screen; or the rays quite at the centre may
be allowed to pass, while others a little farther removed are blocked
off. Stops, each occupying one-eighth of the width, and with centres
situated at the points of trisection, answer well the required
purpose.
It has already been suggested that the principle of energy requires
that the general expression for I^2 in (2) when integrated over the
whole of the plane [xi], [eta] should be equal to A, where A is the
area of the aperture. A general analytical verification has been given
by Sir G. G. Stokes (_Edin. Trans._, 1853, 20, p. 317). Analytically
expressed--
_ _+[oo] _ _
/ / / /
| | I^2 d[xi]d[eta] = | | dxdy = A (9).
_/_/-[oo] _/_/
We have seen that I0^2 (the intensity at the focal point) was equal to
A^2/[lambda]^2f^2. If A' be the area over which the intensity must be I0^2
in order to give the actual total intensity in accordance with
_ _+[oo]
/ /
A'I0^2 = | | I^2 d[xi]d[eta],
_/_/-[oo]
the relation between A and A' is AA' = [lambda]^2f^2. Since A' is in
some sense the area of the diffraction pattern, it may be considered
to be a rough criterion of the definition, and we infer that the
definition of a point depends principally upon the area of the
aperture, and only in a very secondary degree upon the shape when the
area is maintained constant.
4. _Theory of Circular Aperture._--We will now consider the important
case where the form of the aperture is circular.