Microscope, TE>fcESCOPELJ>TERFEROMETER 27
the magnification is proportional to the ratio of the distances from object and image respectively to the center of the lens; hence in the microscope an error in determining the position of the image means a much smaller error in the determination
of the position of the point source. This error could be diminished indefinitely by increasing the magnifying power, were it not for the attendant loss of light and the fact that the light waves, though very minute, are not infinitesimally small. In fact, the same diffraction effects again limit the possibility of indefinite accuracy of measurement. What, then, is the new limit?
Let p, Fig. 22, represent the center of the geometrical
image of a luminous point. This will be a point of maximum brightness, because all parts of the concave wave which converges toward/; reach this point at the same time, and therefore in the same phase. Let us consider an adjacent point q. The parts of the converging wave are no longer at equal
28 Light Waves and Theib Uses
distances from this point, and hence will not arrive in the same phase, and the brightness will be less than at p. At a certain distance pq there will be no light at all. This occurs when the difference of phase between the extreme ray and the central ray is half a wave, that is, calling the wave length Z, when cq — bq = £ I; for these two pairs of rays destroy each other, and the same is true of every two such pairs of rays.
The same is equally true of every point about p at this same distance; hence there will be a dark ring about the bright image. This is succeeded by a bright ring, a second dark ring, and so on.
The radius of the first dark ring may be calculated as follows:
Draw qt at right angles to bp. Then cq — bq = %l. But cq = cp, very nearly, and cp = bp, and bq = &/, so that bp — bq = pt — 11.
But the triangles pqt and pbc are similar, whence pt : pq = be : bp; or, calling r the radius of the first dark ring, F the focal length of the lens, and D the diameter of the lens,
r — — I; that is, the radius of the dark ring is greater than
the length of the light wave, in the same proportion as the focal length of the lens is greater than its diameter.1 For example, if the length of the light wave be taken as one fifty-thousandth of an inch, and the focal length of the lens as one hundred times the diameter, then this radius will be one five-hundredth of an inch — a quantity readily perceptible with a moderate eyepiece. The lack of distinctness of the image would be of the same order, and would be further aggravated by greater magnification, resembling a drawing made with a blunt point.
i Strictly, this is about one-fourth greater on account of the fact that the aperture is circular instead of rectangular.