E T 11 E O K Y 437 extreme rays, and a denote the width of the emergent beam, the dispersion is given by or, if t } 1 o negligible, (8). , The condition of resolution of a double line whose components subtend an angle 6 is that must exceed fa. Hence, in order that a double line may be resolved whose components have indices fji and /J. + S/JL, it is necessary that t should exceed the value given by the following equation :
- = A/8/i ........ (9).
For applications of these results, see SPECTROSCOPE. 14. Theory of Gratings. The general explanation of the mode of action of gratings has been given under LIGHT (vol. xiv. p. 607). If the grating be com posed of alternate transparent and opaque parts, the question may be treated by means of the general integrals (11) by merely limiting the integration to the transparent parts of the aperture. For an investigation upon these lines the reader is referred to Airy s Tracts and to Verdet s Lemons. If, however, we assume the theory of a simple rectangular aperture ( 11), the results of the ruling can be inferred by elementary methods, which are perhaps more instructive. Apart from the ruling, we know that the image of a mathematical line will be a series of narrow bands, of which the central one is by far the brightest. At the middle of this band there is complete agreement of phase among the secondary waves. The dark lines which separate the bands are the places at which the phases of the secondary wave range over an integral number of periods. If now we suppose the aperture AB to be covered by a great number of opaque strips or bars of width d, separated by transparent inter vals of width a, the condition of things in the directions just spoken of is not materially changed. At the central point there is still complete agreement of phase ; but the amplitude is dimin ished in the ratio of a: a + d. In another direction, making a small angle with the last, such that the projection of AB upon it amounts to a few wave-lengths, it is easy to see that the mode of interference is the same as if there were no ruling. For example, when the direction is such that the projection of AB upon it amounts to one wave-length, the elementary components neutralize one another, because their phases are distributed symmetrically, though discontinuously, round the entire period. The only effect of the ruling is to diminish the amplitude in the ratio a : a + d; and, except for the difference in illumination, the appearance of a line of light is the same as if the aperture were perfectly free. The lateral (spectral) images occur in such directions that the projection of the element (a + d) of the grating ipon them is an exact multiple of A. The effect of each of the n elements of the grat ing is then the same ; and, unless this vanishes on account of a parti cular adjustment of the ratio a : d, the resultant amplitude be comes comparatively very great. These directions, in which the retardation between A and B is exactly mn, may be called the principal directions. On either side of any one of them the illum ination is distributed according to the same law as for the central image (m = 0), vanishing, for example, when the retardation amounts to (m?il)A. In considering the relative brightnesses of the different spectra, it is therefore sufficient to attend merely to the principal directions, provided that the whole deviation be not so great that its cosine differs considerably from unity. right- We have now to consider the amplitude due to a single element, ess. which we may conveniently regard as composed of a transparent part a bounded by two opaque parts of width ^d. The phase of the resultant effect is by symmetry that of the component which comes from the middle of a. The fact that the other components have phases differing from this by amounts ranging between amtr/(a + d) causes the resultant amplitude to be less than for the central image (where there is complete phase agreement). If B m denote the brightness of the ? th lateral image, and B that of the central image, we have - cosxdx-z- a + d , a + dj " l (1). If B denote the brightness of the central image when the whole of the space occupied by the grating is transparent, we have and thus B, 1 (2). The sine of an angle can never be greater than unity ; and con sequently under the most favourable circumstances only l/mV* of the original light can be obtained in the m ih spectrum. We con clude that, with a grating composed of transparent and opaque parts, the utmost light obtainable in any one spectrum is in the first, and there amounts to 1/ir 2 , or about -j^, and that for this purpose a and d must be equal. When d = a, the general formula becomes Equal B m :B opaque (3), and trans parent showing that, when m is even, B m vanishes, and that, when m is odd, p ar t s . B m : B = l/7)rir 2 . The third spectrum has thus only ?, of the brilliancy of the first. Another particular case of interest is obtained by supposing a Trans- small relatively to (a + d). Unless the spectrum be of very high parent order, we have simply parts B m :B={a/(a + rf)} 2 (4); small. so that the brightnesses of all the spectra are the same. The light stopped by the opaque parts of the grating, together with that distributed in the central image and lateral spectra, ought to make up the brightness that would be found in the central image, were all the apertures transparent. Thus, if a = d, we should have 112 1 = o + 7 + -s 2 4 IT* which is true by a known theorem. In the general case a Y- 2 " z. 00 1 __ -- , a + d a + d/ ir 2 m =i m- a+dj a formula which may be verified by Fourier s theorem. According to a general principle formulated by Babinet, the Babinet s brightness of a lateral spectrum is not affected by an inter- principle, change of the transparent and opaque parts of the grating. The vibrations corresponding to the two parts are precisely antagonistic, since if both were operative the resultant would be zero. So far as the application to gratings is concerned, the same conclusion may be derived from (2). From the value of B, n : B we see that no lateral spectrum can sur pass the central image in brightness ; but this result depends upon the hypothesis that the ruling acts by opacity, which is generally very far from being the case in practice. In an engraved glass grat- Gratings ing there is no opaque material present by which light could be acting by absorbed, and the effect depends upon a difference of retardation in retarda- passing the alternate parts. It is possible to prepare gratings which tion. give a lateral spectrum brighter than the central image, and the ex planation is easy. For if the alternate parts were equal and_ alike transparent, but so constituted as to give a relative retardation of 4, it is evident that the central image would be entirely extin guished, while the first spectrum would be four times as bright as if the alternate parts were opaque. If it were possible to introduce at every part of the aperture of the grating an arbitrary retardation, all the light might be concentrated in any desired spectrum. By The supposing the retardation to vary uniformly and continuously we whole fall upon the case of an ordinary prism ; but there is then no diffrac- light in tion spectrum in the usual sense. To obtain such it would be neces- O ne sary that the retardation should gradually alter by a wave-length spectrum, in passing over any element of the grating, and then fall back to its previous value, thus springing suddenly over a wave-length. It is not likely that such a result will ever be fully attained in practice ; but the case is worth stating, in order to show that there is no theoretical limit to the concentration of light of assigned wave-length in one spectrum, and as illustrating the frequently observed unsymmetrical character of the spectra on the two sides of the central image. 1 We have hitherto supposed that the light is inci dent perpendicularly upon the grating ; but the theory is easily extended. If the incident rays make an angle with the normal (fig. 9), and the diffracted rays make an angle <j> (upon the same side), the rela tive retardation from each element of width (a + d) to the next is (a + d) (sin + sin $); and this is the quantity which is to be equated to m. Thus sin0 + sin0 = 2sii4(0 + 0).cosi(0-0) = 7nA/(a + d) (5). The "deviation" is (&+<}>}, and is therefore a minimum when = </>, i.e., when the grating is so situated that the angles of inci dence and diffraction are equal. In the case of a reflexion grating the same method applies. 6 and <f> denote the angles with the normal made by the incident grating, and diffracted rays, the formula (5) still holds, and, if the deviation be reckoned from _ the direction of the regularly reflected rays, it is expressed as before by (0 + <t>), and is a minimum when 6 = <f>, that is, when the diffracted rays re turn upon the course of the incident rays. In either case (as also with a prism) the posi tion of minimum deviation leaves the width of the beam unaltered, i.e., neither magnifies nor diminishes the angular width of the object under view. From (5) we see that, when the light falls perpendicularly upon a "radii" (6 = 0), there is no spectrum formed (the image corrc- " ^ Oblique incidence. 1 Phil. May., xlvii. 193, 1874.
A.