Page:Encyclopædia Britannica, Ninth Edition, v. 17.djvu/871

OPTICS 807 disks would be just in contact. Under these conditions there can be no doubt that the star would appear to be fairly resolved, since the brightness of the external ring systems is too small to produce any material confusion, unless indeed the components are of very unequal magni tude. The diminution of star disks with increasing aper ture was observed by W. Herschel ; and in 1823 Fraun- hofer 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 Dawes. The resolving power of telescopes was investigated also by 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 tele scopes of the best construction the performance is not sensibly prejudiced by outstanding aberration, and the limit imposed by the finiteness of the waves of light is practically reached. Verdet has compared Foucault s re sults with theory, and has drawn the conclusion that the radius of the visible part of the image of a luminous point was nearly equal to half the radius of the first dark ring. The theory of resolving power is rather simpler when the aperture is rectangular instead of circular, and when the subject of examination consists of two or more light or dark lines parallel to one of the sides of the aperture. Supposing this side to be vertical, we may say that the definition, or resolving power, is independent of the vertical aperture, and that a double line will be about on the point of resolution when its components subtend an angle eqtial to that subtended by the wave-length of light at a distance equal to the horizontal aperture. The resolving power of a telescope with a circular or rectangular aperture is easily investigated experimentally. The best object is a grating of fine wires, about fifty to the inch, backed by a soda-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 measure ment of the maximum distance the least angle between con secutive 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. In the case of the microscope the wave -theory shows that there must be an absolute limit to resolving power independent of the construction of the instrument. No optical contrivances can decide whether light comes from one point or from another if the distance between them do not exceed a small fraction of the Avave-length. This idea, which appears to have been familiar to Fraunhofer, has recently been expanded by Abbe and Helmholtz into a systematic theory of the microscopic limit. See MICRO SCOPE. Similar principles may be applied to investigate the resolving power of spectroscopes, whether dispersing or diffracting. Consider for simplicity any combination of prisms, anyhow disposed, but consisting of one kind of glass. Let a be the final width and ^ the index of a parallel beam passing through, and let the thicknesses of glass traversed by the extreme rays on either side be t., and ^. It is not difficult to see that, if the index be changed to ft + 5ft, the rays will be turned through an angle given by a Now, if the two kinds of light correspond to a double line which the instrument can just resolve, we have = X/a, and thus < 2 - j = ldfj,, a formula of capital importance in the theory of the dispersing spectroscope. In a well -constructed instrument, t v the smaller thickness traversed, may be small or negligible, and then we may state the law in the following form : the smallest thickness of prisms necessary for the resolution of a double line whose indices are p. and /j. + Sfj- is found by dividing the wave-length by Sfj.. As an example, let it be required to find the smallest thickness of a prism of Chance s "extra dense flint," necessary for resolution of the soda-lines. By Cauchy s formula for the relation between p. a::l X we have M = A+BX- 2 , S/j.= -2BX- 3 SX. From the results given by Hopkinson for this kind of glass we find B= 984xlO- 10 , the unit of length being the centimetre. For the two soda-lines X = 5 889xlO- 5 , 5X=-006xlO~ 5 ; ajul thus the thickness t necessary to resolve the lines is t = X 4 10 10 4 = 1 02 centimetre, 2B5X T9685X" the meaning of which is that the soda-lines .will be resolved if, and will not be resolved unless, the difference of thicknesses of glass traversed by the two sides of the beam amount to one centimetre. In the most favourable arrangement the centimetre is the length of the base of the prism. It is to be understood, of course, that the magnifying power applied is sufficient to narrow the beam ultimately to the diameter of the pupil of the eye ; otherwise the full width would not be utilized. The theory of the resolving power of a diffracting spectroscope, or grating, is even simpler. Whatever may be the position of the grating, a double line of wave-lengths X and X + 5 will be just resolved provided 5X 1 A ~~ mn where n is the total number of lines in the grating, and in is the order of the spectrum under examination. If a grating giving a spectrum of the first order and a prism of extra dense glass have equal power in the region of the soda-lines, the former must have about as many thousand lines as the latter has centimetres of available thickness. The dispersion produced by a grating situated in a given manner is readily inferred from the resolving power. If a be the width of the beam after leaving the grating, the angle 80, corresponding to the limit of resolution, is /a, and thus 50 _mn ~6~~a~ Thus the dispersion depends only upon the order of the spectrum, the total number of lines, and the width of the emergent beam. An obvious inference from the necessary imperfection of optical images is the uselessness of attempting anything like an absolute destruction of aberration. In an instrument free from aberration the waves arrive at the focal point in the same phase. It will suffice for practical purposes if the error of phase nowhere exceeds |X. This corresponds to an error of X in a reflecting and X in a (glass) refracting surface, the incidence in both cases being per pendicular. If we inquire what is the greatest admissible longitudinal aberra tion in an object-glass according to the above rule, we find df=a~ , a being the angular semi-aperture. In the case of a single lens of glass with the most favourable curvatures, 5f is about equal to fa- ; so that a 4 must not exceed X//. For a lens of 3-feet focus this condition is satisfied if the aperture do not exceed 2 inches. When parallel rays fall directly upon a spherical mirror the longitudinal aberration is only about one-eighth as great as for tin; most favourable-shaped single lens of equal focal length and aper ture. Hence a spherical mirror of 3-feet focus might have an aper ture of 2i inches, and the image would not suffer materially from aberration. 1 On general optics the treatises most accessible to the English reader are Parkinson s Optics (3d ed., 1870) and Glazebrook s Physical Optics (1S8:3). Verdet s Lecons d optique physique is an excellent work. Every student should read the earlier parts of Newton s Optics, in which are described the funda mental experiments upon the decomposition of white light. (R.) 1 For fuller information on the subject of the preceding paragraphs see Lord Rayleigh s papers entitled "Investigations in Optics," Phil. Mag., 1879, 1880.