Chapter XVI: Part 16

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.

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.

Writing for brevity

k[xi]/f = p, k[eta]/f = q, (1),

we have for the general expression (S 11) of the intensity

[lambda]^2f^2I^2 = S^2 + C^2 (2),

where
_ _
/ /
S = | | sin(px + qy)dx dy, (3),
_/_/
_ _
/ /
C = | | cos(px + qy)dx dy, (4).
_/_/

When, as in the application to rectangular or circular apertures, the
form is symmetrical with respect to the axes both of x and y, S = 0,
and C reduces to
_ _
/ /
C = | | cos px cos qy dx dy, (5).
_/_/

In the case of the circular aperture the distribution of light is of
course symmetrical with respect to the focal point p = 0, q = 0; and C
is a function of p and q only through [sqrt](p^2 + q^2). It is thus
sufficient to determine the intensity along the axis of p. Putting q =
0, we get
_ _ _+R
/ / / /
C = | | cos px dx dy = 2 | cos px \/(R^2 - x^2) dx,
_/_/ _/-R

R being the radius of the aperture. This integral is the Bessel's
function of order unity, defined by

_[pi]
z /
J1(z) = ---- | cos(z cos [phi]) sin^2 [phi] d[phi] (6).
[pi] _/0

Thus, if x = R cos [phi],

2J1(pR)
C = [pi]^2R ------- (7);
pR

and the illumination at distance r from the focal point is

/ 2[pi]Rr \
4J1^2( --------- )
[pi]^2R^4 \f[lambda]/
I^2 = ----------- . ----------------- (8).
[lambda]^2f^2 / 2[pi]Rr \^2
( --------- )
\f[lambda]/

The ascending series for J1(z), used by Sir G. B. Airy (_Camb.
Trans._, 1834) in his original investigation of the diffraction of a
circular object-glass, and readily obtained from (6), is

z z^3 z^5 z^7
J1(z) = - - ----- + --------- - ------------- + ... (9).
2 2^2.4 2^2.4^2.6 2^2.4^2.6^2.8

When z is great, we may employ the semi-convergent series
_
/ / 2 \ | 3.5.1 /1\^2
J1(z) = / ( ----- ) sin (z - 1/4[pi]) |1 + ------ ( - )
\/ \[pi]z/ |_ 8.16 \z/
_
3.5.7.9.1.3.5 /1\^4 |
- ------------- ( - ) + ... |
8.16.24.32 \z/ _|
_
/ / 2 \ | 3 1 3.5.7.1.3 /1\ ^3
+ / ( ----- ) cos (z - 1/4[pi]) | - . - - --------- ( - )
\/ \[pi]z/ |_8 z 8.16.24 \z/
_
3.5.7.9.11.1.3.5.7 /1\^5 |
+ ------------------ ( - ) - ... | ... (10).
8.16.24.32.40 \z/ _|

A table of the values of 2z^-1J1(z) has been given by E. C. J. Lommel
(_Schlomilch_, 1870, 15, p. 166), to whom is due the first systematic
application of Bessel's functions to the diffraction integrals.

The illumination vanishes in correspondence with the roots of the
equation J1(z) = 0. If these be called z1 z2, z3, ... the radii of the
dark rings in the diffraction pattern are

f[lambda]z1 f[lambda]z2
-----------, -----------, ...
2[pi]R 2[pi]R

being thus _inversely_ proportional to R.

The integrations may also be effected by means of polar co-ordinates,
taking first the integration with respect to [phi] so as to obtain the
result for an infinitely thin annular aperture. Thus, if

x = [rho] cos [phi], y = [rho] sin [phi],

_ _ _R _2[pi]
/ / / /
C = | | cos px dx dy = | | cos (p[rho] cos [theta]) [rho]d[rho] d[theta].
_/_/ _/0 _/0

Now by definition

_1/2[pi]
2 / z^2 z^4 z^6
J0(z) = ---- | cos(z cos[theta])d[theta] = --- + ------- - ----------- + ... (11).
[pi] _/0 2^2 2^2.4^2 2^2.4^2.6^2

The value of C for an annular aperture of radius r and width dr is thus

dC = 2 [pi]J0 (p[rho]) [rho] d[rho], (12).

For the complete circle,

_ pR
2[pi] /
C = ----- | J0(z) zdz
p^2 _/0

2[pi] /p^2R^2 p^4 R^4 p^6 R^6 \
= ------ ( ------ - ------- + ----------- - ... )
p^2 \ 2 2^2.4^2 2^2.4^2.6^2 /

2J1(pR)
= [pi]R^2 . ------- as before.
pR

In these expressions we are to replace p by k[xi]/f, or rather, since
the diffraction pattern is symmetrical, by kr/f, where r is the
distance of any point in the focal plane from the centre of the
system.

The roots of J0(z) after the first may be found from

z .050561 .053041 .262051
---- = i - .25 + ------- - ---------- + ---------- ... (13),
[pi] 4i - 1 (4i - 1)^3 (4i - 1)^5

and those of J1(z) from

z .151982 .015399 .245835
---- = i + .25 - ------- + ---------- + ---------- ... (14),
[pi] 4i + 1 (4i + 1)^3 (4i + 1)^5

formulae derived by Stokes (_Camb. Trans._, 1850, vol. ix.) from the
descending series.[1] The following table gives the actual values:--

+---+--------------------+--------------------+
| | z | z |
| i | ---- for J0(z) = 0 | ---- for J1(z) = 0 |
| | [pi] | [pi] |
+---+--------------------+--------------------+
| 1 | 7655 | 1 2197 |
| 2 | 1 7571 | 2 2330 |
| 3 | 2 7546 | 3 2383 |
| 4 | 3 7534 | 4 2411 |
| 5 | 4 7527 | 5 2428 |
| 6 | 5 7522 | 6 2439 |
| 7 | 6 7519 | 7 2448 |
| 8 | 7 7516 | 8 2454 |
| 9 | 8 7514 | 9 2459 |
|10 | 9 7513 | 10 2463 |
+---+--------------------+--------------------+

In both cases the image of a mathematical point is thus a symmetrical
ring system. The greatest brightness is at the centre, where

dC = 2[pi][rho] d[rho], C = [pi]R^2.

For a certain distance outwards this remains sensibly unimpaired and
then gradually diminishes to zero, as the secondary waves become
discrepant in phase. The subsequent revivals of brightness forming the
bright rings are necessarily of inferior brilliancy as compared with
the central disk.

The first dark ring in the diffraction pattern of the complete
circular aperture occurs when

r/f = 1.2197 X [lambda]/2R (15).

We may compare this with the corresponding result for a rectangular
aperture of width a,

[xi]/f =[lambda]/a;

and it appears that in consequence of the preponderance of the central
parts, the compensation in the case of the circle does not set in at
so small an obliquity as when the circle is replaced by a rectangular
aperture, whose side is equal to the diameter of the circle.

Again, if we compare the complete circle with a narrow annular
aperture of the same radius, we see that in the latter case the first
dark ring occurs at a much smaller obliquity, viz.

r/f = .7655 X [lambda]/2R.

It has been found by Sir William Herschel and others that the
definition of a telescope is often improved by stopping off a part of
the central area of the object-glass; but the advantage to be obtained
in this way is in no case great, and anything like a reduction of the
aperture to a narrow annulus is attended by a development of the
external luminous rings sufficient to outweigh any improvement due to
the diminished diameter of the central area.[2]

The maximum brightnesses and the places at which they occur are easily
determined with the aid of certain properties of the Bessel's
functions. It is known (see SPHERICAL HARMONICS) that

J0'(z) = -J1(z), (16);

1
J2(z) = - J1(z) - J1'(z) (17);
z

2
J0(z) + J2(z) = - J1(z) (18).
z

The maxima of C occur when

d /J1(z)\ J1'(z) J1(z)
-- (-------) = ------ - ----- = 0;
dz \ z / z z^2

or by 17 when J2(z) = 0. When z has one of the values thus determined,

2
- J1(z) = J0(z).
z

The accompanying table is given by Lommel, in which the first column
gives the roots of J2(z) = 0, and the second and third columns the
corresponding values of the functions specified. If appears that the
maximum brightness in the first ring is only about 1/57 of the
brightness at the centre.

+-------------------------------------------+
| z 2z^-1 J1(z) 4z^-2 J1^2(z) |
+-------------------------------------------+
| |
| .000000 +1.000000 1.000000 |
| 5.135630 - .132279 .017498 |
| 8.417236 + .064482 .004158 |
| 11.619857 - .040008 .001601 |
| 14.795938 + .027919 .000779 |
| 17.959820 - .020905 .000437 |
+-------------------------------------------+

We will now investigate the total illumination distributed over the
area of the circle of radius r. We have

[pi]^2R^4 4J1^2(z)
I^2 = ------------- . ------- (19),
[lambda]^2f^2 z^2

where

z = 2[pi]Rr/[lambda]f (20).

Thus
_ _ _
/ [lambda]^2f^2 / /
2[pi] | I^2rdr = ------------- | I^2zdz = [pi]R^2.2 | z^-1 J1^2(z)dz.
_/ 2[pi]R^2 _/ _/

Now by (17), (18)

z^-1 J1(z) = J0(z) - J1'(z);

so that

d d
z^-1J1^2(z) = 1/2 -- J0^2 - 1/2 -- J1^2(z),
dz dz

and

_z
/
2 | z^-1 J1^2(z)dz = 1 - J0^2(z) - J1^2(z) (21).
_/0

If r, or z, be infinite, J0(z), J1(z) vanish, and the whole
illumination is expressed by [pi]R^2, in accordance with the general
principle. In any case the proportion of the whole illumination to be
found outside the circle of radius r is given by

J0^2(z) + J1^2(z).

For the dark rings J1(z) = 0; so that the fraction of illumination
outside any dark ring is simply J0^2(z). Thus for the first, second,
third and fourth dark rings we get respectively .161, .090, .062,
.047, showing that more than 9/10ths of the whole light is
concentrated within the area of the second dark ring (_Phil. Mag._,
1881).

When z is great, the descending series (10) gives

2J1(z) 2 / / 2 \
------ = - / ( ----- ) sin(z - 1/4[pi]) (22);
z z \/ \[pi]z/

so that the places of maxima and minima occur at equal intervals.

The mean brightness varies as z^-3 (or as r^-3), and the integral
found by multiplying it by zdz and integrating between 0 and [oo]
converges.

It may be instructive to contrast this with the case of an infinitely
narrow annular aperture, where the brightness is proportional to
J0^2(z). When z is great,

/ 2
J0(z) = \ / ----- cos(z^-1/4 [pi]).
\/ [pi]z

The mean brightness varies as z^-1; and the integral
_
/ [oo]
| J0^2(z)z dz is not convergent.
_/ 0

5. _Resolving Power of Telescopes._--The efficiency of a telescope is of course intimately connected with the size of the disk by which it represents a mathematical point. In estimating theoretically the resolving power on a double star we have to consider the illumination of the field due to the superposition of the two independent images. If the angular interval between the components of a double star were equal to twice that expressed in equation (15) above, the central disks of the diffraction patterns would be just in contact. Under these conditions there is no doubt that the star would appear to be fairly resolved, since the brightness of its external ring system is too small to produce any material confusion, unless indeed the components are of very unequal magnitude. The diminution of the star disks with increasing aperture was observed by Sir William Herschel, and in 1823 Fraunhofer formulated the law of inverse proportionality. In investigations extending over a long series of years, the advantage of a large aperture in separating the components of close double stars was fully examined by W. R. Dawes.

The resolving power of telescopes was investigated also by J. B. L. Foucault, who employed a scale of equal bright and dark alternate parts; it was found to be proportional to the aperture and independent of the focal length. In telescopes of the best construction and of moderate aperture the performance is not sensibly prejudiced by outstanding aberration, and the limit imposed by the finiteness of the waves of light is practically reached. M. E. Verdet has compared Foucault's results with theory, and has drawn the conclusion that the radius of the visible part of the image of a luminous point was equal to half the radius of the first dark ring.

The application, unaccountably long delayed, of this principle to the microscope by H. L. F. Helmholtz in 1871 is the foundation of the important doctrine of the _microscopic limit_. It is true that in 1823 Fraunhofer, inspired by his observations upon gratings, had very nearly hit the mark.[3] And a little before Helmholtz, E. Abbe published a somewhat more complete investigation, also founded upon the phenomena presented by gratings. But although the argument from gratings is instructive and convenient in some respects, its use has tended to obscure the essential unity of the principle of the limit of resolution whether applied to telescopes or microscopes.

In fig. 4, AB represents the axis of an optical instrument (telescope
or microscope), A being a point of the object and B a point of the
image. By the operation of the object-glass LL' all the rays issuing
from A arrive in the same phase at B. Thus if A be self-luminous, the
illumination is a maximum at B, where all the secondary waves agree in
phase. B is in fact the centre of the diffraction disk which
constitutes the image of A. At neighbouring points the illumination is
less, in consequence of the discrepancies of phase which there enter.
In like manner if we take a neighbouring point P, also self-luminous,
in the plane of the object, the waves which issue from it will arrive
at B with phases no longer absolutely concordant, and the discrepancy
of phase will increase as the interval AP increases. When the
interval is very small the discrepancy, though mathematically
existent, produces no practical effect; and the illumination at B due
to P is as important as that due to A, the intensities of the two
luminous sources being supposed equal. Under these conditions it is
clear that A and P are not separated in the image. The question is to
what amount must the distance AP be increased in order that the
difference of situation may make itself felt in the image. This is
necessarily a question of degree; but it does not require detailed
calculations in order to show that the discrepancy first becomes
conspicuous when the phases corresponding to the various secondary
waves which travel from P to B range over a complete period. The
illumination at B due to P then becomes comparatively small, indeed
for some forms of aperture evanescent. The extreme discrepancy is that
between the waves which travel through the outermost parts of the
object-glass at L and L'; so that if we adopt the above standard of
resolution, the question is where must P be situated in order that the
relative retardation of the rays PL and PL' may on their arrival at B
amount to a wave-length ([lambda]). In virtue of the general law that
the reduced optical path is stationary in value, this retardation may
be calculated without allowance for the different paths pursued on the
farther side of L, L', so that the value is simply PL - PL'. Now since
AP is very small, AL' - PL' = AP sin [alpha], where [alpha] is the
angular semi-aperture L'AB. In like manner PL - AL has the same value,
so that

PL - PL' = 2AP sin [alpha].

According to the standard adopted, the condition of resolution is
therefore that AP, or [epsilon], should exceed 1/2[lambda]/sin [alpha].
If [epsilon] be less than this, the images overlap too much; while if
[epsilon] greatly exceed the above value the images become
unnecessarily separated.

In the above argument the whole space between the object and the lens
is supposed to be occupied by matter of one refractive index, and
[lambda] represents the wave-length _in this medium_ of the kind of
light employed. If the restriction as to uniformity be violated, what
we have ultimately to deal with is the wave-length in the medium
immediately surrounding the object.

Calling the refractive index [mu], we have as the critical value of
[epsilon],

[epsilon] = 1/2[lambda]0/[mu] sin[alpha], (1),

[lambda]0 being the wave-length _in vacuo_. The denominator [mu] sin
[alpha] is the quantity well known (after Abbe) as the "numerical
aperture."

The extreme value possible for [alpha] is a right angle, so that for
the microscopic limit we have

[epsilon] = 1/2[lambda]0/[mu] (2).

The limit can be depressed only by a diminution in [lambda]0, such as
photography makes possible, or by an increase in [mu], the refractive
index of the medium in which the object is situated.

The statement of the law of resolving power has been made in a form
appropriate to the microscope, but it admits also of immediate
application to the telescope. If 2R be the diameter of the
object-glass and D the distance of the object, the angle subtended by
AP is [epsilon]/D, and the angular resolving power is given by

[lambda]/2D sin[alpha] = [lambda]/2R (3).

This method of derivation (substantially due to Helmholtz) makes it
obvious that there is no essential difference of principle between the
two cases, although the results are conveniently stated in different
forms. In the case of the telescope we have to deal with a linear
measure of aperture and an angular limit of resolution, whereas in the
case of the microscope the limit of resolution is linear, and it is
expressed in terms of angular aperture.

It must be understood that the above argument distinctly assumes that
the different parts of the object are self-luminous, or at least that
the light proceeding from the various points is without phase
relations. As has been emphasized by G. J. Stoney, the restriction is
often, perhaps usually, violated in the microscope. A different
treatment is then necessary, and for some of the problems which arise
under this head the method of Abbe is convenient.

The importance of the general conclusions above formulated, as
imposing a limit upon our powers of direct observation, can hardly be
overestimated; but there has been in some quarters a tendency to
ascribe to it a more precise character than it can bear, or even to
mistake its meaning altogether. A few words of further explanation may
therefore be desirable. The first point to be emphasized is that
nothing whatever is said as to the smallness of a single object that
may be made visible. The eye, unaided or armed with a telescope, is
able to see, as points of light, stars subtending no sensible angle.
The visibility of a star is a question of brightness simply, and has
nothing to do with resolving power. The latter element enters only
when it is a question of recognizing the duplicity of a double star,
or of distinguishing detail upon the surface of a planet. So in the
microscope there is nothing except lack of light to hinder the
visibility of an object however small. But if its dimensions be much
less than the half wave-length, it can only be seen as a whole, and
its parts cannot be distinctly separated, although in cases near the
border line some inference may be possible, founded upon experience of
what appearances are presented in various cases. Interesting
observations upon particles, _ultra-microscopic_ in the above sense,
have been recorded by H. F. W. Siedentopf and R. A. Zsigmondy
(_Drude's Ann._, 1903, 10, p. 1).

In a somewhat similar way a dark linear interruption in a bright
ground may be visible, although its actual width is much inferior to
the half wave-length. In illustration of this fact a simple experiment
may be mentioned. In front of the naked eye was held a piece of copper
foil perforated by a fine needle hole. Observed through this the
structure of some wire gauze just disappeared at a distance from the
eye equal to 17 in., the gauze containing 46 meshes to the inch. On
the other hand, a single wire 0.034 in. in diameter remained fairly
visible up to a distance of 20 ft. The ratio between the limiting
angles subtended by the periodic structure of the gauze and the
diameter of the wire was (.022/.034) X (240/17) = 9.1. For further
information upon this subject reference may be made to _Phil. Mag._,
1896, 42, p. 167; _Journ. R. Micr. Soc._, 1903, p. 447.

6. _Coronas or Glories._--The results of the theory of the diffraction patterns due to circular apertures admit of an interesting application to _coronas_, such as are often seen encircling the sun and moon. They are due to the interposition of small spherules of water, which act the part of diffracting obstacles. In order to the formation of a well-defined corona it is essential that the particles be exclusively, or preponderatingly, of one size.

If the origin of light be treated as infinitely small, and be seen in
focus, whether with the naked eye or with the aid of a telescope, the
whole of the light in the absence of obstacles would be concentrated
in the immediate neighbourhood of the focus. At other parts of the
field the effect is the same, in accordance with the principle known
as Babinet's, whether the imaginary screen in front of the
object-glass is generally transparent but studded with a number of
opaque circular disks, or is generally opaque but perforated with
corresponding apertures. Since at these points the resultant due to
the whole aperture is zero, any two portions into which the whole may
be divided must give equal and opposite resultants. Consider now the
light diffracted in a direction many times more oblique than any with
which we should be concerned, were the whole aperture uninterrupted,
and take first the effect of a single small aperture. The light in the
proposed direction is that determined by the size of the small
aperture in accordance with the laws already investigated, and its
phase depends upon the position of the aperture. If we take a
direction such that the light (of given wave-length) from a single
aperture vanishes, the evanescence continues even when the whole
series of apertures is brought into contemplation. Hence, whatever
else may happen, there must be a system of dark rings formed, the same
as from a single small aperture. In directions other than these it is
a more delicate question how the partial effects should be compounded.
If we make the extreme suppositions of an infinitely small source and
absolutely homogeneous light, there is no escape from the conclusion
that the light in a definite direction is arbitrary, that is,
dependent upon the chance distribution of apertures. If, however, as
in practice, the light be heterogeneous, the source of finite area,
the obstacles in motion, and the discrimination of different
directions imperfect, we are concerned merely with the mean brightness
found by varying the arbitrary phase-relations, and this is obtained
by simply multiplying the brightness due to a single aperture by the
number of apertures (n) (see INTERFERENCE OF LIGHT, S 4). The
diffraction pattern is therefore that due to a single aperture, merely
brightened n times.

In his experiments upon this subject Fraunhofer employed plates of
glass dusted over with lycopodium, or studded with small metallic
disks of uniform size; and he found that the diameters of the rings
were proportional to the length of the waves and inversely as the
diameter of the disks.

In another respect the observations of Fraunhofer appear at first
sight to be in disaccord with theory; for his measures of the
diameters of the red rings, visible when white light was employed,
correspond with the law applicable to dark rings, and not to the
different law applicable to the luminous maxima. Verdet has, however,
pointed out that the observation in this form is essentially different
from that in which homogeneous red light is employed, and that the
position of the red rings would correspond to the _absence_ of
blue-green light rather than to the greatest abundance of red light.
Verdet's own observations, conducted with great care, fully confirm
this view, and exhibit a complete agreement with theory.

By measurements of coronas it is possible to infer the size of the
particles to which they are due, an application of considerable
interest in the case of natural coronas--the general rule being the
larger the corona the smaller the water spherules. Young employed this
method not only to determine the diameters of cloud particles (e.g.
1/1000 in.), but also those of fibrous material, for which the theory
is analogous. His instrument was called the _eriometer_ (see
"Chromatics," vol. iii. of supp. to _Ency. Brit._, 1817).

7. _Influence of Aberration. Optical Power of Instruments._--Our investigations and estimates of resolving power have thus far proceeded upon the supposition that there are no optical imperfections, whether of the nature of a regular aberration or dependent upon irregularities of material and workmanship. In practice there will always be a certain aberration or error of phase, which we may also regard as the deviation of the actual wave-surface from its intended position. In general, we may say that aberration is unimportant when it nowhere (or at any rate over a relatively small area only) exceeds a small fraction of the wave-length ([lamda]). Thus in estimating the intensity at a focal point, where, in the absence of aberration, all the secondary waves would have exactly the same phase, we see that an aberration nowhere exceeding 1/4[lambda] can have but little effect.

The only case in which the influence of small aberration upon the
entire image has been calculated (_Phil. Mag._, 1879) is that of a
rectangular aperture, traversed by a cylindrical wave with aberration
equal to cx^3. The aberration is here unsymmetrical, the wave being in
advance of its proper place in one half of the aperture, but behind in
the other half. No terms in x or x^2 need be considered. The first
would correspond to a general turning of the beam; and the second
would imply imperfect focusing of the central parts. The effect of
aberration may be considered in two ways. We may suppose the aperture
(a) constant, and inquire into the operation of an increasing
aberration; or we may take a given value of c (i.e. a given
wave-surface) and examine the effect of a varying aperture. The
results in the second case show that an increase of aperture up to
that corresponding to an extreme aberration of half a period has no
ill effect upon the central band (S 3), but it increases unduly the
intensity of one of the neighbouring lateral bands; and the practical
conclusion is that the best results will be obtained from an aperture
giving an extreme aberration of from a quarter to half a period, and
that with an increased aperture aberration is not so much a direct
cause of deterioration as an obstacle to the attainment of that
improved definition which should accompany the increase of aperture.

If, on the other hand, we suppose the aperture given, we find that
aberration begins to be distinctly mischievous when it amounts to
about a quarter period, i.e. when the wave-surface deviates at each
end by a quarter wave-length from the true plane.

As an application of this result, let us investigate what amount of
temperature disturbance in the tube of a telescope may be expected to
impair definition. According to J. B. Biot and F. J. D. Arago, the
index [mu] for air at t deg. C. and at atmospheric pressure is given by

.00029
[mu] - 1 = -----------.
1 + .0037 t

If we take 0 deg. C. as standard temperature,

[delta][mu] = -1.1 X 10^-6.

Thus, on the supposition that the irregularity of temperature t
extends through a length l, and produces an acceleration of a quarter
of a wave-length,

1/4[lambda] = 1.1 lt X 10^-6;

or, if we take [lambda] = 5.3 X 10^-5,

lt = 12,

the unit of length being the centimetre.

We may infer that, in the case of a telescope tube 12 cm. long, a
stratum of air heated 1 deg. C. lying along the top of the tube, and
occupying a moderate fraction of the whole volume, would produce a not
insensible effect. If the change of temperature progressed uniformly
from one side to the other, the result would be a lateral displacement
of the image without loss of definition; but in general both effects
would be observable. In longer tubes a similar disturbance would be
caused by a proportionally less difference of temperature. S. P.
Langley has proposed to obviate such ill-effects by stirring the air
included within a telescope tube. It has long been known that the
definition of a carbon bisulphide prism may be much improved by a
vigorous shaking.

We will now consider the application of the principle to the formation
of images, unassisted by reflection or refraction (_Phil. Mag._,
1881). The function of a lens in forming an image is to compensate by
its variable thickness the differences of phase which would otherwise
exist between secondary waves arriving at the focal point from various
parts of the aperture. If we suppose the diameter of the lens to be
given (2R), and its focal length f gradually to increase, the original
differences of phase at the image of an infinitely distant luminous
point diminish without limit. When f attains a certain value, say f1,
the extreme error of phase to be compensated falls to 1/4[lambda]. But,
as we have seen, such an error of phase causes no sensible
deterioration in the definition; so that from this point onwards the
lens is useless, as only improving an image already sensibly as
perfect as the aperture admits of. Throughout the operation of
increasing the focal length, the resolving power of the instrument,
which depends only upon the aperture, remains unchanged; and we thus
arrive at the rather startling conclusion that a telescope of any
degree of resolving power might be constructed without an
object-glass, if only there were no limit to the admissible focal
length. This last proviso, however, as we shall see, takes away almost
all practical importance from the proposition.

To get an idea of the magnitudes of the quantities involved, let us
take the case of an aperture of 1/5 in., about that of the pupil of
the eye. The distance f1, which the actual focal length must exceed,
is given by

/
\/ (f1^2 + R^2) - f1 = 1/4[lambda];

so that

f1 = 2R^2/[lambda] (1).

Thus, if [lambda] = 1/4000, R = 1/10, we find

f1 = 800 inches.

The image of the sun thrown upon a screen at a distance exceeding 66
ft., through a hole 1/5 in. in diameter, is therefore at least as well
defined as that seen direct.

As the minimum focal length increases with the square of the aperture,
a quite impracticable distance would be required to rival the
resolving power of a modern telescope. Even for an aperture of 4 in.,
f1 would have to be 5 miles.

A similar argument may be applied to find at what point an achromatic
lens becomes sensibly superior to a single one. The question is
whether, when the adjustment of focus is correct for the central rays
of the spectrum, the error of phase for the most extreme rays (which
it is necessary to consider) amounts to a quarter of a wave-length. If
not, the substitution of an achromatic lens will be of no advantage.
Calculation shows that, if the aperture be 1/5 in., an achromatic lens
has no sensible advantage if the focal length be greater than about 11
in. If we suppose the focal length to be 66 ft., a single lens is
practically perfect up to an aperture of 1.7 in.

Another obvious inference from the necessary imperfection of optical
images is the uselessness of attempting anything like an absolute
destruction of spherical aberration. An admissible error of phase of
1/4[lambda] will correspond to an error of 1/8[lambda] in a reflecting
and 1/2[lambda] in a (glass) refracting surface, the incidence in both
cases being perpendicular. If we inquire what is the greatest
admissible longitudinal aberration ([delta]f) in an object-glass
according to the above rule, we find

[delta]f = [lambda][alpha]^-2 (2),

[alpha] being the angular semi-aperture.

In the case of a single lens of glass with the most favourable
curvatures, [delta]f is about equal to [alpha]^2f, so that [alpha]^4
must not exceed [lambda]/f. For a lens of 3 ft. focus this condition
is satisfied if the aperture does not exceed 2 in.

When parallel rays fall directly upon a spherical mirror the
longitudinal aberration is only about one-eighth as great as for the
most favourably shaped single lens of equal focal length and aperture.
Hence a spherical mirror of 3 ft. focus might have an aperture of 21/2
in., and the image would not suffer materially from aberration.

On the same principle we may estimate the least visible displacement
of the eye-piece of a telescope focused upon a distant object, a
question of interest in connexion with range-finders. It appears
(_Phil. Mag._, 1885, 20, p. 354) that a displacement [delta]f from the
true focus will not sensibly impair definition, provided

[delta]f < f^2[lambda]/R^2 (3),

2R being the diameter of aperture. The linear accuracy required is
thus a function of the _ratio_ of aperture to focal length. The
formula agrees well with experiment.

The principle gives an instantaneous solution of the question of the
ultimate optical efficiency in the method of "mirror-reading," as
commonly practised in various physical observations. A rotation by
which one edge of the mirror advances 1/4[lambda] (while the other edge
retreats to a like amount) introduces a phase-discrepancy of a whole
period where before the rotation there was complete agreement. A
rotation of this amount should therefore be easily visible, but the
limits of resolving power are being approached; and the conclusion is
independent of the focal length of the mirror, and of the employment
of a telescope, provided of course that the reflected image is seen in
focus, and that the full width of the mirror is utilized.

A comparison with the method of a material pointer, attached to the
parts whose rotation is under observation, and viewed through a
microscope, is of interest. The limiting efficiency of the microscope
is attained when the angular aperture amounts to 180 deg.; and it is
evident that a lateral displacement of the point under observation
through 1/2[lambda] entails (at the old image) a phase-discrepancy of a
whole period, one extreme ray being accelerated and the other retarded
by half that amount. We may infer that the limits of efficiency in the
two methods are the same when the length of the pointer is equal to
the width of the mirror.

[Illustraton: FIG. 5.]

We have seen that in perpendicular reflection a surface error not
exceeding 1/8[lambda] may be admissible. In the case of oblique
reflection at an angle [phi], the error of retardation due to an
elevation BD (fig. 5) is

QQ' - QS = BD sec [phi](1 - cos SQQ') = BD sec [phi] (1 + cos 2[phi]) = 2BD cos [phi];

from which it follows that an error of given magnitude in the figure
of a surface is less important in oblique than in perpendicular
reflection. It must, however, be borne in mind that errors can
sometimes be compensated by altering adjustments. If a surface
intended to be flat is affected with a slight general curvature, a
remedy may be found in an alteration of focus, and the remedy is the
less complete as the reflection is more oblique.

The formula expressing the optical power of prismatic spectroscopes
may readily be investigated upon the principles of the wave theory.
Let A0B0 be a plane wave-surface of the light before it falls upon the
prisms, AB the corresponding wave-surface for a particular part of the
spectrum after the light has passed the prisms, or after it has passed
the eye-piece of the observing telescope. The path of a ray from the
wave-surface A0B0 to A or B is determined by the condition that the
optical distance, [int] [mu]ds, is a minimum; and, as AB is by
supposition a wave-surface, this optical distance is the same for both
points. Thus
_ _
/ /
| [mu]ds (for A) = | [mu]ds (for B) (4).
_/ _/

We have now to consider the behaviour of light belonging to a
neighbouring part of the spectrum. The path of a ray from the
wave-surface A0B0 to the point A is changed; but in virtue of the
minimum property the change may be neglected in calculating the
optical distance, as it influences the result by quantities of the
second order only in the changes of refrangibility. Accordingly, the
optical distance from A0B0 to A is represented by [int]([mu] +
[delta][mu])ds, the integration being along the original path A0 ...
A; and similarly the optical distance between A0B0 and B is
represented by [int] ([mu] + [delta][mu])ds, the integration being
along B0 ... B. In virtue of (4) the difference of the optical
distances to A and B is
_ _
/ /
| [delta][mu]ds (along B0 ... B) - | [delta][mu]ds (along A0 ... A) (5).
_/ _/

The new wave-surface is formed in such a position that the optical
distance is constant; and therefore the _dispersion_, or the angle
through which the wave-surface is turned by the change of
refrangibility, is found simply by dividing (5) by the distance AB.
If, as in common flint-glass spectroscopes, there is only one
dispersing substance, [int] [delta][mu] ds = [delta][mu].s, where s is
simply the thickness traversed by the ray. If t2 and t1 be the
thicknesses traversed by the extreme rays, and a denote the width of
the emergent beam, the dispersion [theta] is given by

[theta] = [delta][mu](t2 - t1)/a,

or, if t1 be negligible,

[theta] = [delta][mu]t/a (6).

The condition of resolution of a double line whose components subtend
an angle [theta] is that [theta] must exceed [lambda]/a. Hence, in
order that a double line may be resolved whose components have indices
[mu] and [mu] + [delta][mu], it is necessary that t should exceed the
value given by the following equation:--

t = [lambda]/[delta][mu] (7).

8. _Diffraction Gratings._--Under the heading "Colours of Striated Surfaces," Thomas Young (_Phil. Trans._, 1802) in his usual summary fashion gave a general explanation of these colours, including the law of sines, the striations being supposed to be straight, parallel and equidistant. Later, in his article "Chromatics" in the supplement to the 5th edition of this encyclopaedia, he shows that the colours "lose the mixed character of periodical colours, and resemble much more the ordinary prismatic spectrum, with intervals completely dark interposed," and explains it by the consideration that any phase-difference which may arise at neighbouring striae is multiplied in proportion to the total number of striae.