47T 



UNDULATORY THEORY OF LIGHT. 



UNDULATORY THEORY OF LIGHT. 



478 



of soap- water exhibits, by the gradual subsidence of the fluid, rings of 

 colour exactly conformable to those between glass lenses ; and, before 

 the bubble bursts, a dark spot, about half an inch in diameter, is 

 formed at its upper part. The like phenomena have been observed in 

 thin plates of mica and in bubbles of glass blown so thin as to burst. 



Newton also examined and described the phenomena of the coloured 

 rings or bands between lenses when the light is transmitted through 

 the latter. These rings are less' bright than those which are formed by 

 reflection ; but when the obliquity of the transmitted rays to the 

 plane of the rings is considerable they are sufficiently distinct : in the 

 centre is a white spot, and the colours of all the rings are exactly com- 

 plementary to those of the corresponding rings which are seen by the 

 reflected rays. Newton's arrangement of the coloured tints in the first 

 and second rings or bands, reckoning from the centre, is given in the 

 following table, with the thicknesses of the plates of air, water, and 

 glass, at the places where the tints are produced. The unit of measure 

 is one millionth part of an inch. 



Number of Reflected Transmitted 



the Band. Tints. Tints. Air. Water. date. 



Very black 



Black 



Nearly black 



Blue 



White 



Yellow 



Orange 



Red 



White 



.Yellowish red 

 Black 



Violet 



Blue 

 White 



J 



n 



8 

 9 



H 



6 

 I 



H 

 H 



Pi 



II. Violet White 11 J 6J 71 



Indigo .. 12J 9f 8/j 



Blue Yellow 14 10} 9 



Green Red 14} 11 J 9$ 



Yellow Violet 16$ li| 10} 



Onnge .. 17| 13 llj 



Bright red BUe 18} 13} 11$ 



Scarlet .. 19J 14} 12| 



This table, extended no as to include the seventh band of reflected 

 tints, constitutes that which is called Newton's Scale of Colours. 



It was to explain the phenomena of these coloured rings that Newton 

 proposed the hypothesis of ' Fits of easy Reflection and Transmission,' 

 which in many n/spects explains the laws of the phenomenon. But 

 besides giving no indication beforehand of what ought to be the 

 variation of the diameter of a ring with the refractive index of the 

 interposed medium, or with the obliquity of incidence, it tends to at 

 least one result at variance with observation. According to the hypo- 

 thesis of fits, the central black spot, as well as the dark parts of the 

 rings seen with homogeneous light, ought to be of half the brightness 

 of the brightest parts ; according to the theory of undulations, they 

 ought to be perfectly black. Observation shows, that at least the 

 central spot (which is most easily observed, and does not require homo- 

 geneous light) is perfectly black. 



A general explanation of the rings according to the principles of the 

 undulatory theory may readily be given. Let M x p Q be two plates or 



spherical lenses, which we may suppose plano-convex (though all that 

 ia essential is, that the more curved of the adjacent surfaces should be 

 convex), in contact with each other at 7, ; and let A B be the direction of 

 a pencil, or of a wave of light incident upon the first or upper plate : 

 this will be refracted in some direction as B c, and at the point c part 

 of the pencil will emerge and fall on the other plate in some point E, 

 where it will be partly transmitted through that plate in the direction 

 KB, and partly reflected in the direction EP. Another part of the 

 refracted pencil B c will be reflected at c in the direction c D, some of it 

 emerging at D, and the rest being reflected back into the plate. The 

 reflected pencil E F will also be, in part, transmitted through the upper 

 plate in some direction as p a, and, in port, reflected in the direction 

 F n : at the point H a portion will be transmitted in the direction H 8, 

 while another in reflected in the direction H K, and so on. The two 

 principal reflected pencils are those of which the courses are A B c D and 

 A B c K p o ; and these being each once reflected will be of nearly equal 

 :ty. The other pencils in the general reflected beam, having been 

 reflected 3, 6, 7, 4c. times, will be comparatively weak, except at very 



reat incidences. The two principal transmitted pencils, A B c E R and 

 A B c E F H s will be of very unequal intensity, the latter having been 

 twice reflected, and the former not reflected at all. These will be 

 accompanied by pencils reflected 4, 6, 8, &c. times, which will be com- 

 paratively insignificant. 



Supposing, therefore, for the sake of simplicity, that the light is 

 incident perpendicularly, restricting ourselves to the two most im- 

 portant pencils, and considering first the reflected light, we see that 

 the light reflected from the under sin-face of the thin plate of air has 

 liad to travel a distance 2o in air more than the other stream, D being 

 the distance between the lenses at the point under consideration, or 

 the thickness of the plate of air. We might, therefore, perhaps, 

 expect that the vibrations in the two streams would be in perfect 

 accordance when D was equal to zero, or a multiple of Jx, and in 

 opposition when D was an odd multiple of $A. This would give cor- 

 rectly the law of the variation of the radii of the rings, since D varies 

 as the square of the radius drawn from the point of contact of the 

 lenses, with the single but important exception, that the places of the 

 bright and dark rings are interchanged. But we must remark, that the 

 two reflections take place under opposite circumstances, oue at the 

 surface of a rarer, the other at the surface of a denser medium. Various 

 dynamical analogies, such as the reflection of sound from the end of a 

 tube, according as it is closed or open, the reflection of sound at 

 the common surface of two gases which are supposed not to mix, 

 would make it more probable than the contrary supposition that in one 

 of these two reflections there should be a change of sign, in the other 

 not. A change of sign is equivalent to a change in the length of the 

 path of one of the streams amounting to half an undulation. This 

 change being admitted, theory assigns correctly the law of the variation 

 of the radii of the bright and dark rings. And not only the law of 

 n, but the alaolute mag*it*dt of the rings may be assigned 

 n jiriori, since the length of a wave of light is known by other observa- 

 tions ; and the magnitude so assigned is in conformity with experiment. 

 The explanation of the variation of the scale of the rings with the 

 colour, and of the obliteration of the rings beyond the seventh or 

 thereabouts by overlapping, when the incident light is white, is the same 

 as in other cases of interference. Moreover, if a liquid such as water 

 be interposed between the lenses in place of air, since light travels 

 more slowly in water than in air, in the proportion of 1 to n, the same 

 kind of ring which with air is found at a spot where the distance 

 between the lenses is D, is found with water where it is only D/J.~ I , 

 which explains the law of variation of the radii of the rings with the 

 refractive index of the interposed medium. 



The explanation of the transmitted rings is perfectly similar, the 

 chief differences being, first, that as the interfering streams have been 

 refracted alike, and one of them in addition twice reflected, there is 

 no change of sign, or the interference is determined simply by the 

 ditl'erence of path, without the addition of the half undulation ; and 

 secondly, that the interfering streams are very unequal in intensity, 

 and therefore with homogeneous light the minima are very far from 

 being absolutely black. Thus when light is incident perpendicularly, 

 whether externally or internally, at the common surface of crown 

 glass and ah-, only about the ^.th part of the incident light is refit cted, 

 the remaining pths being transmitted. Hence by two such reflections 

 the intensity is reduced in the proportion of 625 to 1 ; and if we 

 represent by unity the intensity of the light transmitted across the 

 thin plate without reflection, the intensity of the twice reflected light 

 must be represented by j^j. The question may naturally be asked, 

 How can such feeble light by interfering with the former give rise to 

 any sensible rings T The explanation of this paradox is derived from 

 the consideration that in interference we must compound not inim- 

 tilies but vibrations, and thence deduce the intensity by taking the 

 square of the coefficient of vibration. Thus, taking the above numbers 

 as correct, we learn that the coeitieient of vibration will be reduced 

 by one reflection in the proportion of 5 to 1 only, and by two in the 

 proportion of 25 to 1. Hence if we take the coefficient of vibration 

 in the simply transmitted stream as unity, that in the twice reflected 

 stream will be ^, and therefore that in the resultant stream will vary 

 between the limits 1 + .f s , and the intensity will vary between the 

 liinita (1 + ^,) s , or 1 + j" 5 , nearly ; so that the difference between the 

 limits is as much as ^th, or nearly Jth of the mean intensity. 



Next, suppose the light incident obliquely. If /8 be the angle of 

 incidence on the first surface of the upper lens, or which is the same 

 (the lens for this purpose being treated a a plate bounded by parallel 

 surfaces), the angle of refraction into the thin plate of air, it may be 

 shown, as in Airy's ' Tracts ' (' Undulatory Theory,' art. 64), or still 

 more simply by referring everything to sections of the fronts of the 

 waves by the plane of incidence instead of to rays, that the retardation 

 due to the double transit across the plate is 2Dcosj8, in place of 2c, 

 as at a perpendicular incidence. This explains the law of the variation 

 of the rings with the obliquity. 



From the explanation hitherto given it might seem that in the case 

 of the reflected rings the minima, though nearly, ought not to be 

 perfectly black. For the stream reflected at the second surface of the 

 thin plate has to undergo two more refractions than that reflected at 

 the first surface, and at each refraction a small portion of the incident 

 light would be lost to it by reflection. Fresnel first showed (' Annales 

 de Chimie,' torn. 28, p. 129), by taking account of the infinite number 



