IO2 =A2/a2f2 (7).
The formation of a sharp image of the radiant point requires that the illumination become insignificant when t, n 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 X.
When the difference of phase amounts to a, 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 n=o, Elf =sin 0, the phases of the secondary waves range over a complete period when sin 0 = X/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 0 =IX/a, the phases range one and a half2 4. I
periods, and there is revival of illumination. We may compare the brightness with that in the direction 0=o. 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
1 f~'r
3",J cos ¢ d¢: 1,
or 2/3,r:1; so that the brightness in this direction is 4/91r2 of the maximum at 0=0. In like manner we may find the illumination in any other direction, and it is obvious that it vanishes when sin 0 is any multiple of A/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 E=o, the intensity at any point E, n of the diffraction pattern may be represented by
2uaE
f+sinXf. 8
n4 A f 7,2a2e ( ),
~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.
F• t 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 sin2u/u2 from u = o to u=21r. The part corresponding to negative values of.0 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 (cor
responding to B) midway between the two A. 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 E doubtless an observer accustomed to his instrument would recognize the duplicity with certainty. The obliquity, corresponding to u = A, is such that the phases
of the secondary Waves range over a corn
plete o a
period, i.e. such that the projection of FIG. 3.
the horizontal aperture upon this direction
is one wavelength. We conclude that a double line cannot be fairly resolved unless its components subtend an angle exceeding that subtended by the wavelength 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 wavelength 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 objectglass is provided with diaphragms pierced with round holes or slits. One of these, of width equal, say, to onetenth of an inch, is inserted in front of the objectglass, 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
27r
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 wavelengths.
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 2X, instead of a 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, onesixth 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 oneeighth 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 I2 in (2) when integrated over the whole of the plane l;, n 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
ff±~ J2dtdn ffdx'dy=A . . . . (9).
We have seen that Ig (the intensity. at the focal point) was equal to Ai/X2f2. If A' be the area over which the intensity must be 102 in order to give the actual total intensity in accordance with
A'122 =ff + I2dEdn,
the relation between A and A' is AA' =XT. 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.
kt/f=p, kn/f=q' (1),
we have for the general expression (§ Is) of the intensity
?2fuI2=S2+C2 . (2),
where S=ffsin(px+gy)dx dy, . . . (3),
C = ff cos(px+4y) 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=o, and C reduces to
C = ff 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=o, q=o; and C is a function of p and q only through v (p2+g2). It is thus sufficient to determine the intensity along the axis of p. Putting q = o, we get
c ffcos pxdx dy =2f_+:cos I/ (R.2x2) dx,
R being the radius of the aperture. This integral is the Bessel's function of order unity, defined by
Jl(z) _ f "cos (z cos ¢) sin2cp do
0
Thus, if x = R cos
C=ir2R?J1(PR)
p . . (7);1834) in his original investigation of the diffraction of a circular objectglass, and readily obtained from (6), is
z Z3 zo z' (9).
End of Article: IO2 

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