498 
NALORE 
[Marcu 24, 1904 
But now, instead of producing a variable distribution in 
the focal plane of the object glass by means of diaphragms, 
we can do it by means of the diffraction effects of small 
objects on the stage. 
Thus if we put on the stage a grating consisting of a 
series of equidistant spaces, and if e be the grating 
distance, then, taking homogeneous light, a series of 
narrow bands of light, the diffraction images of the source, 
will be produced in the focal plane with darkness between 
them; the central image will be on the axis, and_ if 
0, 4. be the angular distances between the images, 
then sin @,=A/e, sin 0,=2A/e, &c. 
It may be shown that the image in the view plane pro- 
duced by this series of diffracted images is the ordinary 
geometrical image of the grating. It should be observed 
that in this proof there is no discussion of the distribution 
of light in the interspaces between the maxima, and it is 
on this distribution that the question of resolving power 
depends. It is clear, of course, that if we modify the 
number of spectra in the focal plane we modify the image, 
and this is done in an ingenious way in some of the ex- 
periments arranged by Prof. Abbe’s pupils to illustrate the 
theory. 
If we cut out all but the central image the view field 
is uniform, no structure is visible; if we allow the first 
image on either side of the central one to become effective, 
the bands appear in the field in their proper positions, and 
so on. It is said to be the fundamental result of Abbe’s 
theory that the object, the grating, can be fully resolved 
if one diffraction image is formed on either side of the 
central one. It is clear that in this case there will be 
variations of intensity in the view plane; we shall see 
later what they amount to. 
Now the number of spectra is limited by the fact that 
some of the diffracted light may be so obliquely diffracted 
as not to enter the object glass. If 2a be the angular 
aperture of the object glass measured from the axial point 
of the stage, then the nth diffracted image will not appear 
if sin 6, is>sin a, but sin @,=nA/e. 4 
Hence for the nth image to be excluded, nA/e must be 
greater than sin a, but according to Abbe, for resolution 
the first diffracted image must appear, and hence resolu- 
tion is just possible if A/e is equal to sin 8. 
It has been assumed that air is the medium on either 
side of the object glass; if on the object side we have a 
medium of refractive index m, then it is easy to show that 
we must replace sin @ by w sin 0, and the condition of 
resolution is that e should be equal to A/q sin @, or, intro- 
ducing the term numerical aperture for the quantity mw sin 6, 
we have the result that a grating is resolvable if the space 
between the lines is not less than the result found by 
dividing the wave-length of light by the numerical aperture. 
Now, while the truth of this result can in certain cases 
be established, the reasoning given in the books under con- 
sideration is insufficient to prove it. 
In order to decide if the grating can be resolved we must 
establish the law of variation of intensity in the view plane, 
and then consider whether these variations are such that 
they can be detected by the eye. This has been done by 
Lord Rayleigh. The images formed in a microscope are, 
like all other images, produced by interference; in consider- 
ing resolving power we have to consider diffraction effects 
it is true, but the diffraction which concerns us mainly is 
that due to the aperture of the object glass, and only in- 
directly that due to the object viewed. ’ 
Neither is it necessary, if we know completely the dis- 
tribution of the light over the stage, to go back to the 
source in our consideration of the problem; having given 
the distribution over the stage both in amplitude and 
phase, we are potentially able to determine that in the 
yiew plane without reference to the source. Difficulties 
of calculation may stop us, it is true, but that is another 
matter. 
Let us take, again, the case of a grating illuminated by 
plane waves, their plane being parallel to that of the 
grating ; we have to consider the effect due to a series of 
equidistant lines of light; these differ, however, from a 
series of independent equidistant linear sources in that, with 
the grating, the phases of the various sources are the same ; 
we have therefore to remember that interference will take 
NO. 1795, VOL. 69] 
place between the light from the different lines, while with 
a series of independent lines there is no relation between 
the phases; we can calculate the intensity due to each 
source separately, and superpose the whole. 
Lord Rayleigh’s solution of the problem, which is pre- 
sented when a narrow double line in a spectrum is viewed 
through a telescope, or when the attempt is made to resolve 
two close double stars, is better known than his equally 
valid solution of the grating problem, and as it is simpler 
it will be useful to indicate it first. 
The intensity in the view plane for a single linear source, 
assuming for the moment that we are dealing with a tele- 
scope with a rectangular aperture, is given by a certain 
curve. If we assume a second independent source parallel 
to the first we get a similar curve alongside the first. The 
resultant intensity is found by adding the corresponding 
ordinates of the two curves, and the lines will appear as 
double when the drop in the resultant intensity curve is 
sufficient to be detected by the eye. 
Lord Rayleigh suggested that in his case the drop would 
be just distinguishable when the maximum of intensity 
due to the second curve was superposed on the first mini- 
mum due to the first, and experiment has borne this out. 
In this case the two halves of the aperture send light in 
opposite phases to the first minimum, and the angular de- 
flection of the minimum is the angle subtended by the wave- 
length of light at the distance of the breadth of the aper- 
ture. Two lines which subtend a greater angle than this 
can be resolved. 
Similar methods were applied by Lord Rayleigh in 1896 
to the microscope, and additional results have been given 
in his recent communication to the Royal Microscopical 
Society which follows Mr. Gordon’s interesting paper on 
Helmholtz’s theory of resolving power in the Journal of 
the Society. In his paper Mr. Gordon discusses in detail 
Helmholtz’s theory, and points out how far it is from fully 
explaining all the difficulties of microscopic vision. 
In Lord Rayleigh’s earlier paper he deals with (1) two 
independent linear sources viewed through a microscope, 
and shows that they can be resolved if the distance between 
them is half that given by Abbe’s theory; (2) two sources 
which are always in the same phase; in this case resolution 
is impossible if the distance is that given by the theory. 
If, instead of having two sources, either cophasal or 
independent, we have a long series, the problem is more 
complex, but the method is the same. An expression is 
found for the variations of intensity in the view plane, and 
the question is considered whether or no these variations 
are sufficient to be noticed by the eye. 
In the paper the question of the visibility of a dark 
bar on a uniform field is dealt with, and here again a 
distinction must be drawn between the case in which the 
field is self-luminous and that in which it is due to a distant 
source. In the latter case it appears that the image of the 
bar would be marked by a perceptible darkening across the 
field, even when the breadth of the bar was but 1/32 of 
that given by Abbe’s theory, though the breadth of this 
shadow would not be a measure of that of the bar; in the 
former case the fall in intensity over the geometrical image 
is only one-half of what it is in the latter. Moreover, we 
ar2 certain to arrive at erroneous consequences if we apply 
results obtained from the case of a grating of a large 
number of parallel slits to a case such as that of a single 
small aperture through which light is coming or a single 
small obstacle obstructing the light; the diffraction pattern 
due to such an obstacle is entirely different from that due to 
a grating, and the conditions of resolution will be different 
also. 
It appears, then, that while Abbe’s theory of microscopic 
vision is undoubtedly correct in that a small object or objects 
on the stage produce diffraction patterns in the focal plane 
of the object glass, and the illumination in the view plane 
can be inferred from these diffraction images, still this 
method of regarding the question is not the only possible 
one, neither is it necessary to go back to the original source 
if we know the distribution in the object plane. By proceed- 
ing, however, in the way indicated by Lord Rayleigh, we 
can evaluate the distribution of intensity in the view plane, 
at any rate in certain cases, and obtain thus a numerical 
estimate of the resolvability. Reta iGs 
