The Wave Theory of Light

41. From this general expression it is seen that the resultant intensity of the light vibrations is equal to the sum of intensities of the two constituent pencils when they are in perfectagreement and to their difference when they are in exactlyopposite phases, and, lastly, to the square root of the sum of their squares when their phase difference is a quarter of a wave-length, as we’ have already shown. It thus follows that the phase of the wave corresponds ex- actly to the angular position of the resultant of two forces, a and a’. The distance from the first wave to the second is c, to the resultant wave ^-, and from the resultant wave to the/*7T second is c

; accordingly, the corresponding angles are (* C ’ 2TT., i, and 2ir. i. Let us multiply the equation A A + ’ COS ( 2TT =:A COS i by sin t, and the following equation a’ sin ( %TT J =A sin i by cos i. Subtracting one from the other, we have a sin i=a’ sin t 2?r i V which, together with a’ sin ( 27T j =A sin i, gives the following proportion: I 2?r i: sin i : sin 2?r- : : a : a’ : A. 42. The general expression, A sin gyff -Y_t|, for the velocity of the particles in a wave produced by the meeting of two others shows that this wave has the same length as its components and that the velocities at corresponding points are proportional, so that the resultant wave is always of the samenature as its components and differs only in intensity that is to say, in the constant by which we must multiply the velocities in either of the components in order to obtain the correspond107 sn

ing velocities in the resultant. In combining this resultantwith still another new wave., one again arrives at an expressionof the same form a remarkable property of a function of thiskind. Thus in the resultant of any number of trains of wavesof the same length the light particles are always urged by velocities proportional to those of the components at points locatedat the same distance from the end of each wave. [This is seen~by multiplying each of the last three terms in the preceding proportion by sin tat. For then, a sin wt : a

1 sin wi : A sin ut ::a : a’ :A'

: constant ratio. ] APPLICATIONS OF HUYGENS’S PRINCIPLE TO THE PHENOMENAOF DIFFRACTION

43. Having determined the resultant of any number of trainsof light-waves, I shall now show how by the aid of these inter-ference formulae and by the principle of Huygens alone it is possible to explain, and even to compute, all the phenomenaof diffraction. This principle, which I consider as a rigorousdeduction from the basal hypothesis, maybe expressed thus:The vibrations at each point in the wave-front may be consideredas the sum of the elementary motions which at any one instantare sent to that point from all parts of this same wave in anyone of its previous* positions, each of these parts acting inde-pendently the one of the other. It follows from the principleof the superposition of small motions that the vibrations pro-duced at any point in an elastic fluid by several disturbancesare equal to the resultant of all the disturbances reaching thispoint at the same instant from different centres of vibration,whatever be their number, their respective positions, theirnature, or the epoch of the different disturbances. This general principle must apply to all particular cases. I shall suppose that all of these disturbances, infinite in number, are ofthe same kind, that they take place simultaneously, that they*I am here discussing only an infinite train of waves, or the most general vibration of a fluid. It is only in this sense that one can speak of twolight, waves annulling one another when they are half a wave-length apart.The formulae of interference just given do not apply to the case of a sin- gle wave, not ‘o mention the fact that such waves do not occur in nature.

are contiguous and occur in the single plane or on a singlespherical surface. I shall make still another hypothesis withreference to the nature of these disturbances, viz., I shall sup-pose that the velocities impressed upon the particles are all directed in the same sense, perpendicular to the surface of thesphere,* and, besides, that they are proportional to the compression, and in such a/ way that the particles have no retrogrademotion. I have thus reconstructed a primary wave out of partial [secondary] disturbances. We may, therefore, say that thevibrations at each point in the wave-front can be looked uponas the resultant of all the secondary displacements which reachit at the same instant from all parts of this same wave in someprevious position, each of these parts acting independently oneof the other.

44. If the intensity of the primary wave is uniform, it fol- lows from theoretical as well as from all other considerationsthat this uniformity will be maintained throughout its path,provided only that no part of the wave is intercepted or re- tarded with respect to its neighboring parts, because the re- sultant of the secondary displacements mentioned above will be the same at every point. But if a portion of the wave bestopped by the interposition of an opaque body, then the in- tensity of each point varies with its distance from the edge ofthe shadow, and these variations will be especially marked nearthe edge of the geometrical shadow.

Let be the luminous point, AG the screen, AME a wavewhich has just reached A and is partly intercepted by theopaque body. Imagine it to be divided into an infinite number of small arcs Am’, m’m, wM, Mw, nn’, n’n", etc. In orderto determine the intensity at any point P in any of the later positions of the wave BPD, it is necessary to find the resultant of*It is possible for light- waves to occur in which the direction of the ab- solute velocity impressed upon the particles is not perpendicular to the wave surface. In studying the laws of interference of polarized light, I have become convinced since the writing of this memoir that light vibra- tions are at right angles to the rays or parallel to the wave surface. Thearguments and computations contained in this memoir harmonize quite as well with this new hypothesis as with the preceding, because they are quite independent of the actual direction of the vibrations and pre-sup- pose only that the direction of these vibrations is the same for all rays belonging to any system of waves producing fringes.

Fig. 16 all the secondary waves which each of theseportions of the primitive wave would send tothe point P, provided they were acting independently one of the other. Since the impulse communicated to every’part of the primitive wave was directed alongthe normal, the motion which each [part of thewave] tends to impress upon the ether oughtto be more intense in this direction than inany other ; and the rays which would emanatefrom it, if acting alone, would be less and lessintense as they deviated more and more fromthis direction.

45. The investigation of the law according to which theirintensity varies about each centre of disturbance is doubtless avery difficult matter ;* but, fortunately, we have no need ofknowing it, for it is easily seen that the effects produced bythese rays are mutually destructive when their directions aresensibly inclined towards the normal. Consequently, the rayswhich produce any appreciable Qifect upon the quantity oflight received at any point P may be regarded as of equal in-tensity, f Let us now consider the rays EP, FP, and IP, which are sen-* [This is the problem solved by Stokes; Math, and Phys. Papers, vol. ii., p. 243.] f When the centre of disturbance has been compressed, the force of ex-pansion tends to thrust the particles in all directions ; and if they have nobackward motion, the reason is simply that their initial velocities forwarddestroy those which expansion tends to impress upon them towards therear ; but it does not follow that the disturbance can be transmitted onlyalong the direction of the initial velocities, for the force of expansion in aperpendicular direction, for instance, combines with a primitive impulsewithout having its effect diminished. It is clear that the intensity of thewave thus produced must vary greatly at different points of its circumference, not only on account of the initial impulse, but also because the compressions do not obey the same law around the centre of disturbance ; butthe variations of intensity in the resultant wave must follow the law ofcontinuity, and may, therefore, be considered as vanishing throughout asmall angle, especially along the normal to the primitive wave. For theinitial velocities of the particles in any direction whatever are proportional to the cosine of the angle which this direction makes with that of thenormal, so that these components vary much less rapidly than the angleso long as the angle is small. 110

sibly inclined and which meet at P, a point whose distance fromthe wave EA I shall suppose to include a large number of wavelengths. Take the two arcs EF and FI of such a length that thedifferences EP FP and FP IP shall be equal to a half wavelength. Since these rays are quite oblique, and since a halfwave-length is very small compared with their length, thesetwo arcs will be very nearly equal, and the rays which they sendto the point P will be practically parallel ; and since corre-sponding rays on the two arcs differ by half a wave-length, thetwo are mutually destructive. We may then suppose that all the rays which various partsof the primary wave AE send to the point P are of equal in- tensity, since the only rays for which this assumption is notaccurate produce no sensible effect upon the quantity of light which it receives. In the same manner, for the sake of simplifying the calculation of the resultant of all the elementarywaves, we may consider their vibrations as taking place in thesame direction, since the angles which these rays make witheach other are very small ; so that the problem reduces itself to the one which we have already solved namely, to find the resultant of any number ofparallel trains of light-waves of the same length, the intensities and relative positions being given. The intensities are here proportional to the lengths of the il- luminating arcs, and the relative positions of the wave trains are given by the differences of path traversed. 46. Properly speaking, we have considered up to this pointonly the section of the wave made by a plane perpendicular to the edge of the screen projected at A. We shall now considerit in its entirety, and shall think of it as divided by equidistantmeridians perpendicular to the plane of the figure into infinitely thin spindles. We shall then be able to employ the same process of reasoning which we have just used for a section of thewave, and thus show that the rays which are quite oblique are mutually destructive. In the case we are now considering these spindles are indefinitely extended in a direction parallel to the edge of the screen, for the wave is intercepted only on one side. Accordingly theintensity of the resultant of all the vibrations which they sendto the point P would be the same for each of them ; for, owingto the extremely small difference of path, the rays which emanate from these spindles must be considered as of equal inIll

tensity, at least throughout that region of the primitive wavewhich produces a sensible effect upon the light sent to P.Further, it is evident that each elementary resultant will differin phase by the same quantity with respect to the ray comingfrom that point of the spindle nearest P, that is to say, fromthe point at which the spindle cuts the plane of the figure.The intervals between these elementary resultants will then beequal to the difference of path traversed by the rays AP, m’P,mP, etc., all lying in the plane of the figure; and their intensities will be proportional .to the arcs Aw’, m’m, mM., etc. Inorder now to obtain the intensity of the total resultant, wehave to -make the same calculation which we have alreadymade, considering only the section of the wave by a plane per-pendicular to the edge of the screen.* 47. Before deriving the analytical expression for this result-ant I propose to draw from the principle of Huygens some of theinferences which follow from simple geometrical considerations.Let AG represent an opaque body suffi-ciently narrow for one to distinguish fringesin its shadow at the distance AB. Let bethe luminous point and BD be either the fo-/ i i cal pl^ne of the magnifying-glass with whichone observes these fringes or a white cardupon which the fringes are projected. Let us now imagine the original wave di-vided into small arcs Am, mm’, m’m", etc.,Gn, nri, n’n", ri’ri", etc. in such a waythat the rays drawn from the point P in theFi shadow to two consecutive points of divisionwill differ by half a wave-length. All of thesecondary waves sent to the point P by the elements of eachof these arcs will completely interfere with those which emanate* So lone: as the edge of the screen is rectilinear we can determine theposition of the dark and bright bands and their relative intensities by con-sidering only the section of the wave made by a plane which is perpendicular to the edge of the screen. But when the edge of the screen is curvedor composed of straight edges inclined at an angle it is then necessary tointegrate along two directions at right ansrles to each other, or to integratearound the point under consideration. In some particular cases this lattermethod is simpler, as, for instance, when we have to calculate the intensityof the light in the centre of the shadow produced by a screen or in theprojection of a circular aperture.

from the corresponding parts of the two arcs immediately ad-joining it “^ so that, if all these arcs were equal, the rays whichthey would send to the point P would be mutually destructive, with the exception of the extreme arc mA. Half of the in- tensity of this arc would be left, for half the light sent bythe arc mm’ (with which mA is in complete discordance)would be destroyed, by half of the preceding arc m"m’. Assoon as the rays meeting at P are considerably inclined withrespect to the normal, these arcs are practically equal. Theresultant wave, therefore, corresponds in phase almost exactly to the middle of mA, the only arc which producesany sensible effect. It is thus seen that it differs in phaseby one-quarter of a wave-length from the element at the edgeA of the opaque screen. Since the same thing takes place in the other part of the incident wave Qn, the interference be- tween these two vibrations occurring at the point P is deter- mined by the difference of length between the two rays sP andtP, which take their rise at the middle of the arcs Am and Gn,or, what amounts to the same thing, by the difference betweenthe two rays AP and GP coming from the very edge of the opaque body. It thus happens that when the interior fringes under consideration are rather distant from the edges of the geometrical shadow, we are able to apply practically withouterror the formula based upon the hypothesis that the inflected waves have their origin at the very edges of the opaque body;but in proportion as the point P approaches B the arc Ambe- comes greater in comparison with the arc mm’, the arc mm’with respect to the arc m’m”, etc. ; and likewise in the arc mAthe elements in the immediate vicinity of the point A becomesensibly greater than the elements which are situated near the point m, and which correspond to equal differences of path. It happens, therefore, that the effective* ray, sP, will not be the mean between the outside rays, mP and AP, but will morenearly approach the length of the latter. On the other side of the opaque body we have slightly different circumstances. The difference between the ray GP and the effective ray tP ap- proximates more and more nearly a quarter of a wave-length* I have given this name to the distance of the resultant wave from the original wave because the positions of the dark and bright bands are the same as they would be if these effective rays alone produced them. H 113

as the point P moves farther and farther away from D, so thatthe difference of path traversed varies more rapidly betweenthe effective rays sP and tP than between the rays AP and GP; consequently, the fringes in the neighborhood of the point Bought to be a little farther from the centre of the shadow thanwould be indicated by the formula based upon the first hypothesis. 48. Having considered the case of fringes produced by anarrow body, I pass to the consideration of those which arecaused by a small aperture.

Let AG be the aperture through whichthe light passes. I shall at first supposethat it is sufficiently narrow for the darkbands of the first order to fall inside thegeometrical shadow of the screen, andat the same time to be fairly distantfrom the edges B and D.

Let P be thedarkest point in one of these two bands;it is then easily seen that this must cor-respond to a difference of one whole wavelength between the two extreme rays APand GP. Let us now imagine anotherray, PI, drawn in such a way that its length shall be a mean between the othertwo. Then, on account of its marked in- clination to the arc AIG, the point I will fall almost exactly in the middle. We now have the arc di- vided into two parts, whose corresponding elements are almostexactly equal, and send to the point P vibrations in exactlyopposite phases, so that these must annul each other. By the same reasoning it is easily seen that the darkestpoints in the other dark bands also correspond to differencesof an even number of half wave-lengths between ihe two rayswhich come from the edges of the aperture ; and, in like manner, the brightest points of the bright bands correspond todifferences of an uneven number of half wave-lengths that is to say, their positions are exactly reversed as compared withthose which are deduced from the interference of the limitingrays on the hypothesis that these alone are concerned in theproduction of fringes. This is true with the exception of thepoint at the middle, which, on either hypothesis, must be114

bright. The inferences deduced from the theory fringes result from the superposition of all of the disturbancesfrom all parts of the arc AGr are verified by experiments, which at the same time disprove the theory which looks uponthese bands as produced only by rays inflected and reflected at the edges of the diaphragm. These are precisely the phenomena which first led me to recognize the insufficiency of this hypothesis, and suggested the fundamental principle of the theory which I have just explained namely, the principle of Huygens combined with the principle of interference. 49. In the case which we have just considered, where, byvirtue of a very small aperture, the dark bands of the first or- der fall at some distance from the edges of the geometrical shadow, it follows from theory, as well as from experiment, that the distance comprised between the darkest points is al- most exactly double that of the other intervals between the middle points of two consecutive dark bands, and this is all the more nearly true in proportion as the aperture becomessmaller or more distant from the luminous point and from the focus of the magnifying-glass with which one observes the fringes ; for, by sufficiently increasing these distances one mayproduce the same effects with an aperture of any size whatever. But when these distances are not very great, and when their aperture is too large for the rays producing the fringes to be very much inclined to the wave-front, AG-, it follows that corresponding elements of the arcs into which we have supposed a wave to be divided can no longer be considered as each equal to the other, for they are sensibly larger on the side next the band un’der consideration. Under these conditions wecan rigorously deduce the positions of maximum and minimum intensity only by computing the resultant of all the small secondary waves which are sent out by the incident wave.

50. But there is one very remarkable case where a knowledge of this integral is not needed for the determination of the law of the fringes by an aperture of very considerable size. This is the case where a lens is placed in front of the diaphragm, and brings the refracted rays to focus upon the plane in which the fringes are observed. The problem is nowgreatly simplified by the fact that the centre of curvature of the

emergent wave now lies in this plane instead of at the lumi-nous point. Let be the projection of the middlepoint of the aperture upon this plane.From the point as centre, and with aradius equal to AO, let us now describethe arc AI’Gr, which will now representthe incident wave as modified by the inter-position of the lens. If, now, from thepoint P as centre, and with a radius AP,we describe the arc AEF, those portionsof the luminous rays meeting at the pointP which are comprised between the arcAI’G- and the arc AEF will be the differ-ences of path traversed by the secondarywaves ; and, since these two arcs haveequal curvatures and are convex towardsthe same side, it follows that equal differ-ences of path will correspond to equal intervals upon the wavefront AI’G. Let us suppose this wave divided in such a mannerthat any two consecutive rays drawn through the points of di-vision shall differ by one-half a wave-length. If, then, the pointP be located in such a way that the total number of these arcs is even, it will no longer receive any light. For these arcs, takentwo and two, are mutually destructive, since the vibrations dueto corresponding elements are at the same time of equal in-tensity and opposite phase. The light reaching any pointP will be a maximum when the total number of arcs is uneven. The brightest points of the bright bands, therefore, cor-respond to a difference of an uneven number of half wavelengths between the two rays coming from the edges of thediaphragm, and the darkest points on the dark bands to a dif-ference of an even number of half wave-lengths. Consequently,all the dark bands will be equally spaced among themselves,with the exception of the first two, where the interval is ex-actly double that which separates the others. This result,which had already been suggested by theory, I found to bethoroughly confirmed by experiment. I shall cite only oneexperiment of this kind made in homogeneous red light. Inorder to bring the centre of the incident wave to the plane ofthe micrometer wire, I used, instead of an ordinary lens, a116

glass cylinder, which, in order to get the full length of the fringes, I placed with its generating line parallel to the edges of the aperture in the diaphragm. mm. Size of the aperture 2.00 m. Distance from the luminous point to the diaphragm, or a 2 507 Distance from the diaphragm to the micrometer, or b 1.140 mm. Interval between the middle points of the two dark bands of the first order 0.72 Interval between the band of the first order and the third 0.73 Interval between the band of the third order and the fifth 0.72 It will be observed that the first interval is double that of the others. I have observed that the same law holds, even at distances which are not very great, for apertures which are much wider, a centimeter or even a centimeter and a half ; but if we further increase the aperture of the diaphragm, the fringes becomeconfused, however much care be taken to place the micrometer in the focus of the cylindrical lens; which goes to showthat the rays refracted by this glass vibrate in unison [in the same phase} only within rather narrow limits, just as happenswith ordinary lenses. 51. When the aperture of the diaphragm thus backed with a cylindrical lens is not too great, the dark and bright bandsproduced are as sharp as the fringes which result from the union of rays reflected from two mirrors. But, in the latter case, the intensity of the light is the same for all fringes, or, at least, whatever differences there are appear to arise merely fromthe fact that the light employed is not perfectly homogeneous; and if it happen that the bright bands diminish in brilliancy, the dark bands become less dark, so that the sum of the light in one entire fringe remains practically constant. But in theother phenomenon, as one recedes from the centre he observesa rapid diminution of the light, which is easily accounted for by the theory we have just explained. For, indeed, all therays which leave the wave-front AI’Gr and meet at the centreof the bright band of the first order have traversed equalpaths ; so that all the small secondary waves which they bringto this point coincide \in phase] and strengthen each other.

But this is not the case with the other bright bands. Thebrightest band of the second order, for instance, correspondsto a division of the wave AI’G- into three arcs, the extremerays of which differ by one-half a wave-length ; the effectsproduced by two of these arcs annul each other. Consequently, this band receives light from only one-third of the incidentwave-front, while even the effect produced by this third is somewhat diminished by the fact that there is a difference ofone-half a wave-length between the rays from its edges. Asimilar process of reasoning shows that the middle of thebright band of the third order is illuminated by only one-fifthof the wave-front AI’G, the light of this one-fifth being still further diminished by opposition of phase in its extremerays.[Here are omitted six pages, including a geometrical discussionof the general relations between size of aperture (or obstacle), dis- tance of screen, distance of luminous point, etc.] 56. I have just explained the general relations between thesize of any particular fringe and the respective distances of theobstacle from the luminous point and from the micrometer.As we have seen, these laws may be derived from theory quiteindependently of any knowledge of the integral which at eachpoint represents the resultant of all the secondary waves ; butin order to find the absolute size of these fringes, it is essentialthat we compute this resultant, for the positions of maxima andminima of intensity can be determined only by a comparisonof the different values of this resultant, or at least by knowingthe function which represents it. In order to do this, we propose to apply to the principle ofHuygens the method which we have already explained for computing the resultant of any number of trains of waves whentheir intensities and relative positions [phases’] are given.