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See also:J2(z) = ZJI(z)—Ji'(z) . (17); Jo(z)+J2(z) = EJI(z) (18). The See also:maxima of C occur when d (JI(z)1 =Ji'( _Ji(z) =o. dz z z z2 or by (17 when J2(z) =o. When z has one of the values thus determined, .EJ1(z) =Jo(z)• The accompanying table is given by Lommel, in which the first See also:column gives the roots of J2(z) =o, and the second and third columns the corresponding values of the functions specified. If appears that the maximum brightness in the first See also:ring is only about 619 of the brightness at the centre. z 2z-1JI(z) 4z-2Ji2(z) •000000 +1.000000 1.000000 5.135630 - .132279 •017498 8.417236 + 'o64482 •004158 11.619857 - •040oo8 •OOi6oi 14.795938 + •027919 •000779 17.959820 - .020905 .000437 We will now investigate the See also:total See also:illumination distributed over the See also:area of the circle of See also:radius r. We have 12 = r2R°. JZ(z) . . .. (19), z=21rRr/Xf (20). ~• z 21rJ I2rdr=2AR J I2zdz=irR2.2 J z-1J12(z)dz. Now by (17), (18) z1Ji(z) =Jo(z)-JI'(z); so that z-1,112(z) = - 4azJo2(z) -'ia Ji2(z), and 2 fO z--1JI2(z)dz=1—J02(z)—JI2(z) . (21). If r, or z, be See also:infinite, Jo(z), JI(z) vanish, and the whole illumination is expressed by iR2, in accordance with the See also:general principle. In any See also:case the proportion of the whole illumination to be found outside the circle of radius r is given by Jo2(z)+Ji2(z). For the dark rings Ji(z)=o; so that the fraction of illumination outside any dark ring is simply Jo (z). Thus for the first, second, third and See also:fourth dark rings we get respectively •161, .090, •o62, .047, showing that more than the of the whole See also:light is concentrated within the area of the second dark ring (Phil. Meg., 1881). When z is See also:great, the descending See also:series (Io) gives 2Ji(z)= \I (? 1 Z Z 7r2 so that the places of maxima and minima occur at equal intervals. i See also:Airy, loc. cit. " Thus the magnitude of the central spot is diminished, and the brightness of the rings increased, by covering the central parts of the See also:object-See also:glass." The mean brightness varies as z-2 (or as r-2), and the integral found by multiplying it by zdz and integrating between o and 00 converges. It may be instructive to contrast this with the case of an infinitely narrow See also:annular See also:aperture, where the brightness is proportional to Jo2(z). When z is great,
Jo(z) _ .\I(-2 (z iir).
The mean brightness varies as z-4; and the integral f o Jo2(z)zdz is not convergent.
5. Resolving See also:Power of Telescopes.—The efficiency of a See also:telescope is of course intimately connected with the See also:size of the disk by which it represents a mathematical point. In estimating theoretically the resolving power on a See also:double See also:star we have to consider the illumination of the See also: J. B. L. See also:Foucault, who employed a See also:scale of equal See also:bright and dark alternate parts; it was found to be proportional to the aperture and independent of the See also:focal length. In telescopes of the best construction and of moderate aperture the performance is not sensibly prejudiced by outstanding See also: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 See also:drawn the conclusion that the radius of the visible See also:part of the See also:image of a luminous point was equal to See also:half the radius of the first dark ring. The application, unaccountably long delayed, of this principle to the See also:microscope by H. L. F. See also:Helmholtz in 1871 is the See also:foundation of the important See also:doctrine of the microscopic limit. It is true that in 1823 Fraunhofer, inspired by his observations upon gratings, had very nearly See also:hit the See also:mark .2 And a little before Helmholtz, E. See also:Abbe published a somewhat more See also:complete investigation, also founded upon the phenomena presented by gratings. But although the See also: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 See also:resolution whether applied to telescopes or microscopes. In fig. 4, AB represents the See also:axis of an See also:optical See also: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 neigh- FIG. 4. bouring point P, also self- luminous, in the See also: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 2 " See also:Man kann daraus schliessen, was moglicher Weise durch Mikroskope nosh zu sehen ist. Ein mikroskopischer Gegenstand z. B, dessen Durchmesser=(a) ist, and der aus zwei Theilen besteht, kann nicht mehr als aus zwei Theilen bestehend erkannt See also:werden. Dieses zeigt uns See also:erne Grenze See also:des Sehvermogens durch Mikroskope " (See also: In virtue of the general law that the reduced optical path is stationary in value, this retardation may be calculated without See also:allowance for the different paths pursued on the farther See also:side of L, L', so that the value is simply PL—PL'. Now since AP is very small, AL'—PL'= AP See also:sin a, where a is the angular semi-aperture L'AB. In like manner PL—AL has the same value, so that PL—PL'=2AP sin a. According to the standard adopted, the See also:condition of resolution is therefore that AP, or e, should exceed IX/sin a. If a be less than this, the images overlap too much; while if a greatly exceed the above value the images become unnecessarily separated. In the above argument the whole space between the object and the See also:lens is supposed to be occupied by See also:matter of one refractive See also:index, and X represents the wave-length in this See also:medium of the See also:kind of light employed. If the restriction as to uniformity be violated, what we have ultimately to See also:deal with is the wave-length in the medium immediately surrounding the object. Calling the refractive index µ, we have as the See also:critical value of e, e=ZXo/µsin a, (1) , X0 being the wave-length in vacuo. The denominator /s sin a is the quantity well known (after Abbe) as the " numerical aperture." The extreme value possible for a is a right See also:angle, so that for the microscopic limit we have e = P0/µ (2). The limit can be depressed only by a diminution in X0, such as See also:photography makes possible, or by an increase in u, 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 See also:form appropriate to the microscope, but it admits also of immediate application to the telescope. If 2R be the See also:diameter of the object-glass and D the distance of the object, the angle subtended by AP is e/D, and the angular resolving power is given by a/2D sin a= a/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 See also:head the method of Abbe is convenient. The importance of the general conclusions above formulated, as imposing a limit upon our See also:powers of See also:direct observation, can hardly be overestimated; but there has been in some quarters a tendency to ascribe to it a more precise See also:character than it can See also:bear, or even to See also:mistake its meaning altogether. A few words of further explanation may therefore be desirable. The first point to be emphasized is that teething whatever is said as to the smallness of a single object that may be made visible. The See also: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 See also:element enters only when it is a question of recognizing the duplicity of a double star, or of distinguishing detail upon the See also:surface of a See also: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 See also:Fine some inference maybe 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 See also:illustration of this fact a See also:simple experiment may be mentioned. In front of the naked eye was held a piece of See also:copper See also:foil perforated by a fine See also:needle hole. Observed through this the structure of some See also:wire See also:gauze just disappeared at a distance 'from the eye equal to 17 in., the gauze containing 46 meshes to the See also:inch. On the other See also:hand, a single wire 0.034 in. in diameter remained fairly visible up to a distance of 20 ft. The ratio between the limiting angle subtended by the periodic structure of the gauze and the diameter ~f the wire was (.022/.034) X (240/17) =9.1. For further See also:information upon this subject reference may be made to Phil. Mag., 1896, 42, p. 167; Journ. R. Mier. See also: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 See also:sun and See also:moon. They are due to the interposition of small spherules of See also:water, which See also:act the part of diffracting obstacles. In order to the formation of a well-defined See also: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 See also:focus, whether with the naked eye or with the aid of a telescope, the whole of the light in the See also: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 See also: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 See also: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 See also:escape from the conclusion that the light in a definite direction is arbitrary, that is, dependent upon the See also:chance See also:distribution of apertures. If, however, as in practice, the light be heterogeneous, the source of finite area, the obstacles in See also: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, § 4). The diffraction See also: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 See also:lycopodium, or studded with small metallic disks of See also: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 See also:measures of the diameters of the red rings, visible when See also: See also:Influence of Aberration. Optical Power of See also: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 See also:regular aberration or dependent upon irregularities of material and workmanship. In practice there will always be a certain aberration or See also: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 See also:rate over a relatively small area only) exceeds a small fraction of the wave-length (X). 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 ;X can have but little effect. The only case in which the influence of small aberration upon the entire image has been calculated (Phil. Meg., 1879) is that of a rectangular aperture, traversed by a cylindrical wave with aberration equal to cx3. The aberration is here unsymmetrical, the wave being in advance of its proper See also:place in one half of the aperture, but behind in the other half. No terms in x or x2 need be considered. The first would correspond to a general turning of the See also: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) See also: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 See also:ill effect upon the central See also:band (§ 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 See also: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 See also: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 See also:tube of a telescope may be expected to impair definition. According to J. B. See also:Biot and F. J. D. AYago, the indexµ for See also:air at t° C. and at atmospheric pressure is given by 00029 -=1+•UO37 t' If we take o° C. as standard temperature, an -= -1.1 IX lo-it. Thus, on the supposition that the irregularity of temperature t extends through a length 1, and produces an See also:acceleration of a quarter of a wave-length, iX=I.1 ltXIo-4; or, if we take X =5.3 X10-5, 1t=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 I ° C. lying along the See also:top of the tube, and occupying a moderate fraction of the whole See also:volume, would produce a not insensible effect. If the See also: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. See also: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 See also:carbon bisulphide See also: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 See also:refraction (Phil. Meg., 1881). The See also:function of a lens in forming an image is to compensate by its variable thickness the See also:differences of phase which would other-See also:wise 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 See also:original differences of phase at the image of an infinitely distant luminous point diminish without limit. When f attains a certain value, say fl, the extreme error of phase to be compensated falls to }X. 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 See also:idea of the magnitudes of the quantities involved, let us take the case of an aperture of in., about that of the See also:pupil of the eye. The distance f,, which the actual focal length must exceed, is given by See also:R2)—f1=;a; so that f'=2R2/X . Thus, if a = Tan, R =-6, we find fl=800 inches. The image of the sun thrown upon a screen at a distance exceeding 66 ft., through a hole 4 in. in diameter, is therefore at least as well defined as that seen-direct. As the minimum focal length increases with the square of the aperture, a quite impracticable distance would be required to See also:rival the resolving power of a See also:modern telescope. Even for an aperture of 4 in., fl would have to be 5 See also:miles. A similar argument may be applied to find at what point an achromatic lens becomes sensibly See also:superior to a single one. The question is whether, when the See also:adjustment of focus is correct for the central rays of the spectrum, the error of phase for the most extreme rays (which it is necessary to consider) amounts to a quarter of a wave-length. If not, the substitution of an achromatic lens will be of no advantage. Calculation shows that, if the aperture be 4 in., an achromatic lens has no sensible advantage if the focal length be greater than about II in. If we suppose the focal length to be 66 ft., a single lens is practically perfect up to an aperture of 1.7 in. Another obvious inference from the necessary imperfection of optical images is the uselessness of attempting anything like an See also:absolute destruction of spherical aberration. An admissible error of phase of 4X will correspond to an error of 4X in a reflecting and 41 in a (glass) refracting surface, the incidence in both cases being perpendicular. If we inquire what is the greatest admissible See also:longitudinal aberration Of) in an object-glass according to the above rule, we find Sf =X a--2 (2), a being the angular semi-aperture. In the case of a single lens of glass with the most favourable curvatures, Of is about equal to a2f, so that See also:a4 must not exceed X/f. For a lens of 3 ft. focus this condition is satisfied if the aperture does not exceed 2 in. When parallel rays fall directly upon a spherical See also:mirror the longitudinal aberration is only about one-eighth as great as for the most favourably shaped single lens of equal focal length and aperture. Hence a spherical mirror of 3 ft. focus might have an aperture of 21 in., and the image would not suffer materially from aberration. On the same principle we may estimate the least visible displacement of the eye-piece of a telescope focused upon a distant object, a question of interest in connexion with range-finders. It appears (Phil. Mag., 1885, 20, p. 354) that a displacement Sf from the true focus will not sensibly impair definition, provided Sf <f2a/R2 (3), 2R being the diameter of aperture. The linear accuracy required is thus a function of the ratio of aperture to focal length. The See also:formula agrees well with experiment. The principle gives an instantaneous See also:solution of the question of the ultimate optical efficiency in the method of " mirror-See also:reading," as commonly practised in various See also:physical observations. A rotation by which one edge of the mirror advances ;X (while the other edge retreats to a like amount) introduces a phase-discrepancy of a whole period where before the rotation there was complete agreement. A rotation of this amount should therefore be easily visible, but the limits of resolving power are being approached; and the conclusion is independent of the focal length of the mirror, and of the employment of a telescope, provided of course that the reflected image is seen in focus, and that the full width of the mirror is utilized. A comparison with the method of a material pointer, attached to the parts whose rotation is under observation, and viewed through a microscope, is of interest. The limiting efficiency of the microscope is attained when the angular aperture amounts to 180°; and it is evident that a lateral displacement of the point under observation through 4X entails (at the old image) a phase-discrepancy of a whole period, one extreme See also:ray being accelerated and the other re- tarded by half that amount. We may infer that the limits of efficiency in the two methods are the same when the length of the pointer is equal to the width of the mirror. We have seen that in perpendicular reflection a surface error not exceeding }a may be admissible. In the case of oblique reflection at an angle co, the error of retardation due to an See also:elevation BD (fig. 5) is QQ'-QS=BD sec 0(1-cosSQQ') =BD sec 0(1 +See also:cos 20) =2BDcos#; from which it follows that an error of given magnitude in the figure of a surface is less important in oblique than in perpendicular reflection. It must, however, be See also:borne in mind that errors can sometimes be compensated by altering adjustments. If a surface intended to be See also:flat is affected with a slight general curvature, a remedy may be found in an alteration of focus, and the remedy is the less complete as the reflection is more oblique. The formula expressing the optical power of prismatic spectroscopes may readily be investigated upon the principles of the wave theory. Let AoBo be a plane wave-surface of the light before it falls upon the prisms, AB the corresponding wave-surface for a particular part of the spectrum after the light has passed the prisms, or after it has passed the eye-piece of the observing telescope. The path of a ray from the wave-surface AoBo to A or B is determined by the See also:con- dition that the optical distance, f Ads, is a minimum; and, as AB is by supposition a wave-surface, this optical distance is the same for both points. Thus fads (for A) = fads (for B) . . . . (4). We have now to consider the behaviour of light belonging to a neighbouring part of the spectrum. The path of a ray from the wave-surface AoBo to the point A is changed; but in virtue of the minimum See also:property' the change may be neglected in calculating the optical distance,as it influences the result by quantities of the second order only in the changes of refrangibility. Accordingly, the optical distance from AoBo to A is represented by (A+SA)ds, the integration being along the original path Ao . . ; and similarly the optical distance between AoBo and B is represented by f (A+SA)ds, the integration being along Bo B. In virtue of (4) the difference of the optical distances to A and B is f Sµ ds (along Bo : . . B) f Suds (along Ao . . A) (g);
The new wave-surface is formed in such a position that the optical distance is constant; and therefore the See also:dispersion, or the angle through which the wave-surface is turned by the change of refrangibility, is found simply by dividing (5) by the distance AB. If, as in See also:common See also:flint-glass spectroscopes, there is only one dispersing
substance, f Su ds =Sias, where s is simply the thickness traversed by the ray. If t, and tl be the thicknesses traversed by the extreme rays, and a denote the width of the emergent beam, the dispersion o is given by
o = SAt/a (6).
The condition of resolution of a double See also:line whose components subtend an angle B is that 0 must exceed X/a. Hence, in order that a double line may be resolved whose components have indices A and A+SA, it is necessary that t should exceed the value given by the following e q u a t i o n : —
(7)•
8. Diffraction Gratings.—Under the heading "See also:Colours of Striated Surfaces," See also: His See also:recent See also:discovery of the " fixed lines " allowed a precision of observation previously impossible. He constructed gratings up to 340 periods to the inch by straining fine wire over screws. Subsequently he ruled gratings on a layer of See also:gold-See also:leaf attached to glass, or on a layer of grease similarly supported, and again by attacking the glass itself with a See also:diamond point. The best gratings were obtained by the last method, but a suitable diamond point was hard to find, and to preserve. Observing through a telescope with light perpendicularly incident, he showed that the position of any ray was dependent only upon the grating interval, viz. the distance from the centre of one wire or line to the centre of thenext; and not otherwise,upon the thickness of the wire and the magnitude of the interspace. In different gratings the lengths of the spectra and their distances from the axis were inversely proportional to the grating. interval, while with a given grating the distances of the various spectra from the axis were as 1, 2, 3, &c. To Fraunhofer we owe the first accurate measurements of wave-lengths, and the method of separating the overlapping spectra by a prism dispersing in the perpendicular direction. He described also the complicated patterns seen when a point of light is viewed through two superposed gratings, whose lines See also:cross one another perpendicularly or obliquely. The above observations relate to transmitted light, but Fraunhofer extended his inquiry to the light reflected. To eliminate the light returned from the hinder surface of an engraved grating, he covered it with a See also:black See also:varnish. , It then appeared that under certain angles of incidence parts of the resulting spectra were completely polarized. These remarkable researches of Fraunhofer, carried out in the years 1817-1823, are republished in his Collected Writings (See also:Munich, z888). The principle underlying the See also:action of gratings is identical with that discussed in § 2, and exemplified in T. L. Soret's " See also:zone plates.. The alternate Fresnel's zones are blocked out or otherwise modified; in this way the original See also:compensation is upset and a revival of light occurs in unusual directions. If the source be a point or a line, and a collimating lens be used, the incident waves may be regarded as plane. If, further, on leaving the grating the light be received by a focusing lens; e.g. the object-glass of a telescope, the Fresnel's zones are reduced to parallel and equidistant straight strips, which at certain angles coincide with the ruling. The directions of the lateral spectra are such that the passage from one element of the grating to the corresponding point of the next implies a retardation of an integral number of wave-lengths. If the grating be composed of alternate transparent and opaque parts, the question may be treated by means of the general integrals (§ 3) 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, to Verdet's Leona, or to R. W. See also:Wood's Physical See also:Optics. If, however, we assume the theory of a simple rectangular aperture (§ 3) ; 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 See also:middle of this band there is complete agreement of phase among the secondary waves. The dark lines which See also: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 intervals 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 See also:amplitude is diminished in the ratio of a: a+d. In another direction, making a small angle with the last, such that the See also: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, See also: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 See also:appearance of a line of light is the same as if the aperture were perfectly See also:free. The lateral (spectral) images occur in such directions that the projection of the element (a+d). of the grating upon them is an exact multiple of X. The effect of each of the n elements of the grating is then the same; and, unless this vanishes on See also:account of a particular adjustment of the ratio a: d, the resultant amplitude becomes comparatively very great. These directions, in which the retardation between A and B is exactly mnX, may be called the See also:principal directions. On 'either side of any one of them the illumination is distributed according to the same law as for the central image (m = o), vanishing, for example, when the retardation amounts to (mn t i)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. We have now to consider the amplitude due to a single element, which we may conveniently regard as composed of a transparent part a bounded by two opaque parts of width id. 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 aamr/(a+d) causes the resultant amplitude to be less than for the central image (where there is complete phase agreement). = See also:SAN --ti)/a, or, if t1 be negligible, If B. denote the brightness of the mth lateral image, and Bo that of the central image, we have See also:aar a+d 2ama 2 a+d 2. ama B. : Bo = [ am,'" cosx dx _ a+d ] _ (am d) sin 2a+d+d (1). a+d If B denotes the brightness of the central image when the whole of the space occupied by the grating is transparent, we have Bo:B =See also:a2:(a+d)2, and thus 1 Zama B=m2asina+d+d (2). The sine of an angle can never be greater than unity; and consequently under the most favourable circumstances only I/m2a2 of the original light can be obtained in the mt' spectrum. We conclude 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 I/a2, or about , and that for this purpose a and d must be equal. When d=a the general formula becomes B,,, B = si m ma (3) , showing that, when m is even, B. vanishes, and that, when m is See also:odd, B. : B =1 /m2a2. The third spectrum has thus only I of the brilliancy of the first. Another particular case of interest is obtained by supposing a small relatively to (a+d). Unless the spectrum be of very high order, we have simply B = (a/ (a+d) } 2 . . . (4) ; 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 1=2+4+P (i++-...), which is true by a known theorem. In the general case a a 2 2"' ° 1 2 maa a+d V a+d) +~r2, m2sin a+d , a formula which may be verified by See also:Fourier's theorem. According to a general principle formulated by J. Babinet, the brightness of a lateral spectrum is not affected by an interchange 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,,, : B0 we see that no lateral spectrum can surpass the central image in brightness; but this result depends upon the See also:hypothesis that the ruling acts by opacity, which is generally very far from being the case in practice. In an engraved glass grating there is no opaque material See also:present by which light could be absorbed, and the effect depends upon a difference of retardation in passing the alternate parts. It is possible to prepare gratings which 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 4X, it Is evident that the central image would be entirely extinguished, 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 supposing the retardation to vary uniformly and continuously we - , fall upon the case of an ordinary prism : but there is then no diffraction spectrum in the usual sense. To obtain such it would be necessary that the retardation should gradually alter by a wave-length in passing over any element of the grating, and then fall back to its previous value, thus springing suddenly over a wave-length (Phil. Meg., 1874, 47, p. 193). It is not likely that such a result will ever be fully attained in practice; but the case is See also:worth stating, in order to show that there is no theoretical limit to the concentration and as illustrating the frequently observed unsymmetrical character of the spectra on the two sides of the central image.' We have hitherto supposed that the light is incident perpen- ' The last See also:sentence is repeated from the writer's article " Wave Theory " in the 9th edition of this See also:work, but A. A. Michelson's ingenious See also:echelon grating constitutes a realization in an unexpected manner of what was thought to be impracticable.-[R.]dicularly upon the grating; but the theory is easily extended. If the incident rays make an angle 0 with the normal (fig. 6), and the diffracted rays make an angle 4, (upon the same side), the relative retardation from each element of width (a+d) to the next is (a+d) (sin 0+sin 4)) ; and this is the quantity which is to be equated to mX. Thus sin 0+sin 4) =2 sin (0+4)) cos 4(B—¢) =mX/(a+d) (5). The " deviation " is (8+0), and is therefore a minimum when 0 = 4), i.e. when the grating is so situated that the angles of incidence and diffraction are equal. In the case of a reflection grating the same method applies. If B and 4) denote the angles with the normal made by the incident 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+4)), and is a mini-mum when 0=4), that is, when the diffracted rays return upon the course of the incident rays. In either case (as also with a prism) the position 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 grating (0=o), there is no spectrum formed (the image corresponding to m=o not being counted as a spectrum), if the grating interval c or (a+d) is less than X. Under these circumstances, if the material of the grating be completely transparent, the whole of the light must appear in the direct image, and the ruling is not perceptible. From the absence of spectra Fraunhofer argued that there must be a microscopic limit represented by a; and the inference is plausible, to say the least (Phil. Mag., 1886). Fraunhofer should, however, have fixed the microscopic limit at 4X, as appears from (5), when we suppose 0= la, ¢=47. We will now consider the important subject of the resolving power of gratings, as dependent upon the number of lines (n) and the order of the spectrutn observed (m). Let BP (fig. 8) be the direction of the principal maximum (middle of central band) for the wave-length A in the mth spectrum. Then the relative retardation of the extreme rays (corresponding to the edges A, B of the grating) Is mna. If BQ w be the direction for the first minimum (the darkness between the central and first lateral band), the relative retardation of the extreme rays is (mn+i)A. Suppose now that A+SA is the wave-length for which BQ gives the principal maximum, then (mn+1)a=mn(A-f-SA); whence SA/a= I/mn (6). According to our former standard, this gives the smallest difference of wave-lengths in a double line which can be just resolved; and we conclude that the resolving power of a grating depends only upon the total number of lines, and upon the order of the spectrum, without regard to any other considerations. It is here of course assumed that the n lines are really utilized. In the case of the D lines the value of SA/h is about 1/1000; so that to resolve this double line in the first spectrum requires moo lines, in the second spectrum 500, and so on. It is especially to be noticed that the resolving power does not depend directly upon the closeness of the ruling. Let us take the case of a grating 1 in. broad, and containing moo lines, and consider the effect of interpolating an additional moo lines, so as to bisect the former intervals. There will be destruction by interference of the first, third and odd spectra generally; while the advantage gained in the spectra of even order is not in dispersion, nor in resolving power, but simply in brilliancy, which is increased four times. If we now suppose half the grating cut away, so as to leave moo lines in half an inch, the dispersion will not be altered, while the brightness and resolving power are halved. There is clearly no theoretical limit to the resolving power of gratings, even in spectra of given order. But it is possible that, as suggested by See also:Rowland ,2 the structure of natural spectra may be too coarse to give opportunity for resolving powers much higher than those now in use. However this may be, it would always be possible, with the aid of a grating of given resolving power, to construct artificially from white light mixtures of slightly different wave-length whose resolution or otherwise would discriminate between powers inferior and superior to the given one.° 2 Compare also F. F. Lippich, Pogg. Ann. cxxxix. p. 465, 187o; Rayleigh, Nature (See also:October 2, 1873). 2 The power of a grating to construct light of nearly definite wave-length is well illustrated by Young's comparison with the See also:production of a musical See also:note by reflection of a sudden See also:sound from a See also:row of palings. The objection raised by Herschel (Light, § 703) to this comparison depends on a misconception If we define as the " dispersion " in a particular part of the spectrum the ratio of the angular interval dB to the corresponding increment of wave-length dX, we may See also:express it by a very simple formula. For the alteration of wave-length entails, at the two limits of a diffracted wave-front, a relative retardation equal to mndX. Hence, if a be the width of the diffracted beam, and dB the angle through which the wave-front is turned, ad() = mn da, or dispersion = mn/a . . . . (7). The resolving power and the width of the emergent beam See also:fix the optical character of the instrument. The latter element must eventually be decreased until less than the diameter of the pupil of the eye. Hence a wide beam demands treatment with further apparatus (usually a telescope) of high magnifying power. In the above discussion it has been supposed that the ruling is accurate, and we have seen that by increase of m a high resolving power is attainable with a moderate number of lines. But this See also:procedure (apart from the question of illumination) is open to the objection that it makes excessive demands upon accuracy. According to the principle already laid down it can make but little difference in the principal direction corresponding to the first spectrum, provided each line See also:lie within a quarter of an interval (a+d) from its theoretical position. But, to obtain an equally See also:good result in the mth spectrum, the error must be less than rim of the above amount.' There are certain errors of a systematic character which demand See also:special consideration. The spacing is usually effected by means of a See also:screw, to each revolution of which corresponds a large number (e.g. one See also:hundred) of lines. In this way it may happen that although there is almost perfect periodicity with each revolution of the screw after (say) too lines, yet the too lines themselves are not equally spaced. The " ghosts " thus arising were first described by G. H. Quincke (Pogg. Ann., 1872, 146, p. I), and have been elaborately investigated by C. S. See also:Peirce (Ann. Journ. Math., 1879, 2, p• 330), both theoretically and experimentally. The general nature of the effects to be expected in such a case may be made clear by means of an illustration already employed for another purpose. Suppose two similar and accurately ruled transparent gratings to be superposed in such a manner that the lines are parallel. If the one set of lines exactly bisect the intervals between the others, the grating interval is practically halved, and the previously existing spectra of odd order vanish. But a very slight relative displacement will cause the apparition of the odd spectra. In this case there is approximate periodicity in the half interval, but complete periodicity only after the whole interval. The advantage of approximate bisection lies in the superior brilliancy of the surviving spectra; but in any case the See also:compound grating may be considered to be perfect in the longer interval, and the definition is as good as if the bisection were accurate. The effect of a See also:gradual increase in the interval (fig. 9) as we pass across, the grating has been investigated by M. A. See also:Cornu (C.R., 1875, 8o, p. 655), who thus explains an See also:anomaly observed by )lf E. E. N. Mascart. The latter found that certain gratings exercised a converging power upon the spectra formed upon one side, and a corresponding diverging power upon the spectra on the other side. Let us suppose that the light is incident perpendicularly, and that the grating interval increases from the centre towards that edge which lies nearest to the spectrum under observation, and decreases towards the hinder edge. It is evident that the waves from both halves of the grating are ac- celerated in an increasing degree, as we pass from the centre out- pared with the phase they would possess. were the central value of the grating interval maintained throughout. The irregularity of spacing has thus the effect of a See also:convex lens, which accelerates the marginal relatively to the central rays. On the other side the effect is reversed. This kind of irregularity may clearly be present in a ' It must not be supposed that errors of this order of magnitude are unobjectionable in all cases. The position of the middle of the bright band representative of a mathematical line can be fixed with a spider-line See also:micrometer within a small fraction of the width of the band, just as the accuracy of astronomical observations far transcends the separating power of the instrument.degree surpassing the usual limits, without loss of definition, when the telescope is focused so as to secure the best effect. It may be worth while to examine further the other See also:variations from correct ruling which correspond to the various terms expressing the deviation of the wave-surface from a perfect plane. If x and y be co-ordinates in the plane of the wave-surface, the axis of y being parallel to the lines of the grating, and the origin corresponding to the centre of the beam, we may take as an approximate equation to the wave-surface x2 z =2p+Bxy+ZP,+ax3+15'xzy+7xy2+Syi+. . . (8) ; and, as we have just seen, the See also:term in.x-2 corresponds to a linear error in the spacing. In like manner, the term in yz corresponds to a general curvature of the lines (fig. Io), and does not influence the definition at the (See also:primary) focus, although it may introduce astigmatism .2 If we suppose that everything is symmetrical on the two sides of the primary plane y=o, the coefficients B, ,B, S vanish. In spite of any inequality between p and p', the definition will be good to this order of approximation, provided a and y vanish. The former measures the thickness of the priniary focal line, and the latter measures its curvature. The error of ruling giving rise to a is one in which the intervals increase or decrease in both directions from the centre outwards (fig. II), and it may often be compensated by a slight rotation in See also:azimuth of the object-glass of the observing telescope. The term in y corresponds to a variation of curvature in See also:crossing the grating (fig. 12). When the plane zx is not a plane of symmetry, we have to consider the terms in xy, x2y, and y3. The first of these corresponds to a deviation from See also:parallelism, causing the interval to alter gradually as we pass along the lines (fig. 13). The error thus arising may be compensated by a rotation of the object-glass about one of the diameters y =x. The term in x2y corresponds to a deviation from parallelism in the same direction on both sides of the central line (fig. 14) ; and that in y3 would be caused by a curvature such that there is a point of inflection at the middle of each line (fig. 15). All the errors, except that depending on a, and especially those depending on y and S, can be diminished, without loss of resolving power, by contracting the See also:vertical aperture. A linear error in the spacing, and a general curvature of the lines, are eliminated in the ordinary use of a grating. The explanation of the difference of focus upon the two sides as due to unequal spacing was verified by Cornu upon gratings purposely constructed with an increasing interval. He has also shown how to rule a plane surface with lines so disposed that the grating shall of itself give well-focused spectra. A similar idea appears to have guided H. A. Rowland to his brilliant invention of See also:concave gratings, by which spectra can be photographed without any further optical appliance. In these d-instruments the lines are ruled upon a spherical surface of See also:speculum See also:metal, and mark the intersections of the surface by a system of parallel and equidistant planes, o~------of which the middle member passes through Z the centre of the See also:sphere. If we consider for the present only the primary plane of symmetry, the figure is reduced to two dimensions. Let AP (fig. 16) represent the surface of the grating, 0 being the centre of the circle. Then, if Q be any radiant point and Q' its image (primary focus) in the spherical mirror AP, we have 1 I __ 2cos4 v1+u a where vi=AQ', u=AQ, a=OA, ¢=angle of incidence QAO, equal to the angle of reflection Q'AO. If Q be on the circle described upon OA as diameter, so that u= a cos 0, then Q' lies also upon the same circle; and in this case it follows from the symmetry that the unsymmetrical aberration (depending upon a) vanishes. This disposition is adopted in Rowland's instrument; only, in addition to the central image formed at the angle 0'=0, there are a series of spectra with various values of ¢', but all disposed upon the same circle. Rowland's investigation is contained in the See also:paper already referred to; but the following account of the theory is in the form adopted by R. T. Glazebrook (Phil. Mag., 1883). In order to find the difference of optical distances between the courses QAQ', QPQ', we have to express QP-QA, PQ'-AQ'. To find the former, we have, if OAQ = 4, AOP =w, See also:QP2=u2+4a2sin 22w-4au sin Zw sin (zw-O) = (u+a sin 4, sin w)2-a2 sin 2¢sin 2w+4a sin 22w(a-u cos ¢). 2 " In the same way we may conclude that in flat gratings any departure from a' straight line has the effect of causing the dust in the slit and the spectrum to have different,foci—a fact sometimes observed " (Rowland, " On Concave Gratings for Optical Purposes," Phil. Mag., See also:September 1883). 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