prism light angle rays surface glass water homogeneous line found
REFRACTION. - If homogeneous light be refracted at a C plane surface separating two homogeneous isotropic media, I.! the incident and refracted rays are in one plane with the ti normal to the surface, and the sines of their inclinations to it are in a constant ratio.
The law of single refraction was put in a form equivalent to this (all but one word) for the first time by Snell in Leyden, some time before 1626. It was first published in 1637 by Descartes, who undoubtedly obtained it from Snell ; but lie gave it without any mention of its author.
The one word referred to is homogeneous as applied to the incident light. For the fact that white light consists of innumerable different homogeneous constituents, which are separated from one another by refraction, was first established by Newton. We quote his own account of this important discovery from his letter to Oldenburg, dated Feb. , 1671 . - "To perform my late promise to you, I shall without further ceremony acquaint yon, that in the year 1666 (at which time I applied myself to the grinding of optick-glasses of other figures than spherical) I procured me a triangular glass-prism, to try therewith the celebrated phienomena of colours. And in order thereto, having darkened my chamber, a-nd made a small hole in my window-shuts, to let in a convenient quantity of the sun's light, I placed. my prism at its entrance, that it might be thereby refracted to the opposite, wall. It was at first a very pleasing divertisement, to view the vivid and intense colours produced thereby ; but after a while applying myself to consider them more circumspectly, I became surprised to see them in an oblong form ; which, according to the received laws of refractions, I expected should have been circular. They were terminated at the sides with streight lines, but at the ends the decay of light was so gradual, that it was difficult to determine justly what was their figure, yet they seemed semicircular.
" Comparing the length of this coloured spectrum with its breadth, I found it about five times greater ; a disproportion so extravagant, that it excited me to a more than ordinary curiosity of examining from whence it might proceed. I could scarce think, that the various thickness of the glass, or the termination with shadow or darkness, could have any influence on light to produce such an effect : yet I thought it not amiss, first to examine those circumstances, and so tried what would happen by transmitting light through parts of the glass of divers thicknesses, or through holes in the window of divers bignesses, or by setting the prism without, so that the light might pass through it, and be refracted, before it was terminated by the hole : but I found none of those circumstances material. The fashion of the colours was in all these cases the same.
"Then I suspected, whetherhy any unevenness in th e glass, or other contingent irregularity, these colours might be thus dilated. And to try this, I took another prism like the former, and so placed it, that the light passing through them both, might be refracted contrary ways, and so by the latter returned into that course from which the former had diverted it : for by this means I thought the regular effects of the first prism would be destroyed by the second prism, but the irregular ones more augmented, by the multiplicity of refractions. The event was, that the light, which by the first prism was diffused into an oblong form, was by the second reduced into an orbicular one, with as much regularity as when it did not at all pass through them. So that whatever was the cause of that length, it was not any contingent irregularity.
" I then proceeded to examine more critically, what might be effected by the difference of the incidence of rays coining from divers parts of the sail ; and to that end, measured the several lines and angles belonging to the image. Its distance from the hole or prism was 22 foot ; its utmost length 131 inches ; its breadth 2i ; the diameter of the hole 1 of an inch. The angle width the rays, tending towards the middle of the image, made with those lines, in which they would have proceeded without refraction, was 44 deg. 56 min. and the vertical angle of the prism 63 deg. 12 min. Also the refractions on both sides the prism, that is, of the incident and emergent rays, were, as near as I could make them, equal ; and consequently about 54 deg. 4 min. And the rays fell perpendicularly upon the wall. Now subducting the diameter of the hole from the length and breadth of the image, there remains 13 inches in the length, and 28 the breadth, comprehended by those rays which passed through the center of the said hole ; and consequently the angle of the hole, which that breadth subtended, was about 31 min. answerable to the sun's diameter ; but the angle which its length subtend -1, was more than 5 such diameters, namely, 2 deg. 49 min.
"Having made these observations, I first computed from them the refractive power of that glass, and found it measured by the ratio of the sines 20 to 31 ; and then by that ratio I computed the refractions of two rays flowing from opposite parts of the sun's discus, so as to differ 31 min. in their obliquity of incidence, and found that the emergent rays should have comprehended an angle of about 31 min. as they did before they were incident.
" But because this computation was founded on the hypothesis of the proportionality of the shies of incidence and refraction, which though by my own experience I could not imagine to be so erroneous, as to make that angle but 31 min. which in reality was 2 deg. 49 min. yet my curiosity caused me again to take my prism : and having placed it at my window, as before, I observed, that by turning it a little about its axis to and fro, so as to vary its obliquity to the light, more than an angle of 4 or 5 degrees, the colours were not thereby sensibly translated from their place on the wall ; and consequently by that variation of incidence, the quantity of refraction was not sensibly varied. By this experiment, therefore, as well as by the former computation, it was evident, that the difference of the incidence of rays, flowing from divers parts of the sun, could not make them after decussation diverge at a sensibly greater angle, than that at which they before converged ; which being at most but about 31 or 32 min. there still remained some other cause to be found out, from whence it could be 2 deg. 49 min.
"Then I began to suspect, whether the rays, after their trajection through the prism, did not move in curve lines, and according to their more or less curvity, tend to divers parts of the wall. And it increased my suspicion, when I remembered that I had often seen a tennis-ball, struck with an oblique racket, describe such a curve line. For, a circular as well as a progressive motion being communicated to it by that stroke, its parts, on that side where the motions conspire, must press and beat the contiguous air more violently than on the other, and there excite a reluctancy and re-action of the air proportionably greater. And for the same reason, if the rays of light should possibly be globular bodies, and by their oblique passage out of one medium into another acquire a circulating motion; they ought to feel the greater resistance from the ambient nether, on that side where the motions conspire, and thence be continually bowed to the other. But notwithstanding this plausible ground of suspicion, when I came to examine it, I could observe no such curvity in them. And besides (which was enough for my purpose) I observed, that the difference betwixt the length of the image and the diameter of the hole, through which the light was transmitted, was proportionable to their distance.
" The gradual removal of these suspicions at length led me to the experimeittum, erveis, which was this. I took two boards, and placed one of them close behind the prism at the window, so that the light might pass through a small hole, made in it for the purpose, and fall on the other board, which I placed at about 12 feet distance, having first made is small hole in it also for sonic of that incident light to pass through. Then I placed another prism behind this second board, so that the light trajected through both the boards might pass through that also, and be again refracted. before it arrived at the wall. This done, I took the first prism in my hand, and turned it to and fro slowly about its axis, so much as to make the several parts of the image, cast on the second board, successively pass through the hole in it, that I might observe to what places on the wall the second prism would refract them. And I saw, by the variation of those places, that the light, tending to that end of the image towards which the refraction of the first prism was made, did in the second prism suffer a refraction considerably greater than the light tending to the other end. And so the true cause of the length of that image was detected to be no other, than that light is not similar or homogeneal, but consists of di:form rays, some of which are more refrangible than others ; so that without any difference in their incidence on the same medium, some shall be more refracted than others ; and therefore that, according to their particular degrees of refrangibility, they were transmitted through the prism to divers parts of the opposite wall."
The constant ratio mentioned in the above statement of the law of refraction is called the refractive index. Its I numerical value depends upon the nature of the two media, and also upon the quality of the homogeneous light. It is usually greater for orange light than for red, for yellow than for orange, and so on, - so that the violet rays are often called the "more refrangible" rays.' medium, the angle of refraction is usually less than that of incidence.
If the refractive index for a particular kind of light from a medium A into another B be that from B to A is 1 .
In other words, a refracted ray may be sent back by the path by which it came.
If /Li be the refractive index for a particular ray from A into B, and i.e., that for the same ray from A into C, that /42 from B into C is - .
These being premised, let us consider a source 4f homogeneous light in air shining on a surface of water. Here we may take p. as about equal to 1.
Let MN (fig. 12) be the perpendicular to the water surface at the point where the incident ray AP meets it. In the plane APM make the angle QPN such that sin APM=4 sin QPN, then PQ is the refracted ray. If QP be produced back ward; to meet the vertical line BA in q, we may present this statement i:i the form PR ,PB PA = P.,1=1Pq If the rays fall nearly perpendicularly- on the surface, we may put (approximately) B for P, and we have Bq-111:1 .
Hence, an eye placed under water and nearly in the vertical through A, sees a virtual image of A at q, one-third farther from the surface of the water.
As Pis taken farther and farther from A, the angle of incidence becomes more nearly a right angle, and the sine of the angle of refraction becomes more nearly equal to -3. A ray cannot go from air into water so as to make, with tia perpendicular to the surface, an angle whose sine is greatel than :1. The true nature of this curious statement is however, best seen when we suppose the source to bp under water, and the light to be refracted into air. I APQ (fist. 13) be the course of a ray, we have as before Hence, if P, be taken so that re- it is clear that q coincides with D, or the ray AP„ refracte( at P1, runs along the surface of the water. If AP., be les than P211, no point q can he found ; so that the ray AP eq1010t get out of the water. It is found to be completeI reflected in the water. This reflexion unaccompanied b: refraction is called total retlexion. The limiting angle o incidence (at P1) which separates the totally reflected ray from those which (at least partially) escape into air is called the critical angle. When an equilateral triangular c prism of glass is placed in a ray of sunlight, and made to a rotate, we see (besides the spectra formed by refraction) beams of white light reflected alternately from the outside and the inside of each face. The totally reflected ray from the inside is seen to be very much brighter than that reflected from the outside.
To an eye placed nearly in the vertical of A, A appears at A„, where Thus a clear stream, when we look vertically into it, 1 appears to be of only iths of its real depth. But when we look more and more obliquely, seeing A for instance by the i may QP, the image appears nearer and nearer to the surface ; , or, if A be at the bottom, the water will appear more and more shallow ; and all objects in it will appear to be crowded towards the surface. Thus the part of a stick immersed in water appears bent up towards the surface of the water.
Again, to an eye at A, all objects above the water will be seen within a right cone of which AB is the axis and API a side. The rest of the water surface, outside the cone just mentioned, shows us objects at the bottom by reflexion in a perfect mirror.
All this is on the supposition that the light is homogeneous. But when white light is emitted by A, the point A, will be nearer the surface for each constituent the greater is the refractive index. Thus a white point at A will appear drawn out into a coloured line whose lower end is red and upper end violet.
It is easily seen from the law of refraction that light, on passing through a plate of homogeneous material with parallel faces, finally emerges in a direction parallel to that at incidence, and therefore white light comes out from it still white. If the plate be water in a glass vessel with parallel sides, a body placed close to one side, while the eye is close to the other, appears to be at iths of its real distance from the eye.
The explanation of the law of refraction, on the corpus- cular theory, was given by Newton. It is still of importance, as the earliest instance of the solution of a problem involving molecular forces. Newton shows that, as the molecular forces on a corpuscle balance one another at every point inside either of the media, its velocity must be constant in each, but that in passing through the surface of separation of the two media the square of the velocity perpendicular to the surface undergoes a finite change.
Thus, if v be the velocity in air, a the angle of incidence, then in glass the velocity parallel to the surface is still v sin a, but that perpendicular to the surface is 1/2cos2a+a2. Thus the whole velocity is VI:2+a'; and, if a' be the angle of refraction, v- + a- sin a =v sin a.
Prisms.--When the surfaces are plane, but not parallel, we have what is called a prism.
The general nature of the action of a prism will be easily understood by the help of the previous illustrations, if we restrict ourselves to the case of a prism of very small angle and to rays passing nearly perpendicular to each of its faces. Thus, the rays falling nearly at right angles to its surface from a point A (fig. 14) will, after the first refraction, appear to diverge from a luminous line RV, red at the end next to A, violet at the other. This line is in the perpendicular AB from A to the first surface of the prism. Draw from R. and V perpendiculars RS, VT to the second surface of the prism. Join BS, BT, and draw A7', Av parallel to them so as to cut ItS in r and VT in v. To an eye behind the prism, the bright point A will appear to be drawn out into a coloured line IT, red at the end nearest to A.
If A be a n trrow bright line of light, parallel to the edge of the prism, it will appear to be drawn out into a rectangle consisting of images of the line ranged parallel to one another, and due to the various homogeneous constituents of white light in order of their refrangibility. If the light do not contain rays of every degree of refrangibility, some of these images will be wanting, and there will be corre sponding dark lines or bands crossing this spectrum (es it is called). The amount by which any part of this spectrum is shifted from the true position of the bright slit depends (cateris paribus) upon the amount of the refraction. It also depends on the angle of the prism. And, for a given angle, the length of the spectrum depends upon the difference between the refractive indices for the red and the violet rays. This is called the dispersion.
If a second prism, of the same glass, and of the same angle, as the first, be placed in a reversed position behind it (as indicated by the dotted lines in the figure), the effect of the two would be simply that of a plate of glass with parallel faces ; the emergent rays would each be parallel to its original direction, and there would be no separation of colours. The reversed prism would therefore undo the work of the direct prism. Then we should have no dispersion, but we should also have no refraction. We have, however, as has already been shown, an increase of divergence, i.e., the image is nearer to the eye than the object. Blair, Brewster, and Amici devised combinations of two pairs of prisms of the same glass, those of each pair having their edges parallel, such that the combination acted as a sort of achromatic telescope of low power.
Newton, from some rough experiments, hastily concluded that the amount of dispersion is in all substances proportional to that of the refraction. If such were the ease it is easy to see that prisms of two differently refracting materials and of correspondingly different angles, combined (as above described) so as to annul the dispersion, would likewise annul the refraction. Thus Newton was led to suppose that refraction without dispersion is impossible.
It was found by Hall in 1733, and afterwards (independently) by Dollond, that this idea is incorrect - that, in fact, we have in certain media large refraction with com paratively small dispersion, and vice versa, and thus that the dispersion may be got rid of while a part of the refraction remains. James Gregory had previously conjectured that this might be done by using, as is done in the eye, more media than one. Thus we have for certain specimens of flint and crown glass, whose optical constants were carefully measured by Fraunhofer, the following values of the refractive index for three definite kinds of homogeneous light : - spectively, and are given off by incandescent hydrogen. D is the orange-yellow ray of sodium.
When the angle of the prism is very small (the only case we treat here), we may consider An, as approximately a straight line, whose length is (ea,teris paribus) proportional to the angle of the prism. Also the distances Ar, Ar, are to one another in the proportion of the refractive indices of red and violet rays, each diminished by unity. Hence, for a prism of flint glass such as was employed by Fraunhofer, the distances from A to its images formed by these three kinds of homogeneous light respectively are very nearly as When a prism of crown glass is used they are nearly as „ crown glass to or as 2 : 1. Hence if we make the small angle of the crown-glass prism twice that of the flint, and observe A through the two prisms, with their edges turned in opposite directions, the C and F images will coincide. Both, however, will be displaced fro:n the real direction of A as if a prism had been employed, with its edge turned as that of the crown glass was, and to the same extent as that prism would have displaced them had its refractive index been about 1'21 and the same for the two kinds of light C and F.
In fact, the displacements by the flint prism are as This combination of prisms is called achromatic, or colourless, but is not perfectly so. For if we inquire into the displacement of the D image, we find that it is as for the flint prism ; but as in the opposite direction, for the crown prism. Hence its whole displacement is as 425, a little greater than that common to C and F. The reason for this is obvious from Fraunhofer's numbers given above. The interval from C to D is to that from 0 to F in a greater ratio in crown than in flint glass, - so that the spectra given by these media are not similar. The rays of higher refrangibility are more separated in proportion to those of lower refrangibility in flint than in crown glass. This is the irrationality of dispersion - which, so far as we yet know, renders absolute achromatism unattainable. Three lenses in combination give a better attempt at achromatism than can be made with two ; and some remarkable results were obtained by Blair,' with two glass lenses enclosing a lenticular portion of a liquid.
By looking through a prism at a very narrow slit, formed by the window shutters of a darkened room, 1rVollaston L. (in 1802) found that the light of the sky (i.e., sunlight) lives a spectrum which is not continuous. It is crossed by dark bands, as already hinted. These bands are due to the deficiency of intensity of certain definite kinds of homogeneous light. They were, independently, rediscovered, and their positions measured, by Fraunhofer2 in 1817 with far inure perfect optical apparatus. He also found similar, but not the same, deficiencies in the light from various fixed stars. The origin of these bands will be explained in RADIATION, along with the theory of their application in spectrum analysis. In optics they are useful to an extreme degree in enabling us to measure refractive indices with very great precision. Wollaston's own account of his discovery is as follows :- " If a beam of day-light be admitted Into a dark room by a crevice iyth of an inch broad, and received by the eye at the distance of 10 or 12 feet, through a prism of flint-glass, free from veins, held near the eye, the begin is seen to be separated into the four following colours only, red, yellowish-green, blue, and violet, in the proportions represented in fig "The line A that bounds the red side of the spectrum is somewhat confused, which seems in part owing to want of power in the eye to converge red light. The line B, between red and green, in a certain position of the prism is perfectly distinct ; so also are D and E, the two limits of violet. But C, the limit of green and blue, is not so clearly marked as the rest ; and there are also on each side of this limit other distinct dark lines./ and g, either of which in an imperfect experiment might be mistaken for the boundary of these colours.
"The position of the prism in which the colours are most clearly divided is when the incident light makes about equal angles with two of its sides. I then found that the spaces AB, BC, CD, DE occupied by them were nearly as the numbers 16, 23, 36, 25."3 The mode of formation of a spectrum which was employed by Newton, and which is still used when the spectrum is to be seen by many spectators at a time, differs from that just explained in this, that the light from a source A is allowed to pass through the prism, and to fall on a white screen at a considerable distance from it. In this case the paths of the various rays as they ultimately escape from the prism are found by joining the points 7,, with the prism and producing these lines to meet the screen. Unless one surface of the prism be • covered by an opaque plate, with a narrow slit in it parallel to the edge of the prism, the spectrum produced in this way is very impure, i.e., the spaces occupied by the various homogeneous rays overlap one another. To make it really pure an achromatic lens is absolutely requisite. This leads us, naturally, to the consideration of the refraction of light at spherical surfaces.
Refraction at a Spherical Surface. - Following almost exactly the same course as that taken with reflexion above, let 0 (fig. 15) be the centre of curvature of the spherical refracting surface AB. Let U be the point-source of homogeneous light, and let PV be the prolongation (backwards) of the path pursued, after refraction, by the ray UP.
Trans. RAE., vol. iii. (1791).
B, f, C, p, D, E, . . . Wollaston's lines.
D, b, F, U, H, . . Frannhofer's lines.
There is no single line in Fraunhofer's drawing of the spectrum, nor is there any in the real spectrum, coincident with the line C of Wollaston's, and indeed lie himself describes it as not being so clearly marked as the rest.' I have found, however, that this line C corresponds to a number of lines half-way between b and F, which, owing to the absorption of the atmosphere, arc particularly visible in the light of the sky near the horizon." - Brewster, Report on Optics, Brit. Association, 1832.
Then, rigorously, we have sin 1.11'0 - ,u sin 01'V, whereµ is the index of refraction between the two media employed. This may be written (by omitting a common factor) as