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.
[Illustration: Fig. 4.]
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).