Interference_Of_Light

Interference Of Light. § 1. This term1 and the ideas underlying it were introduced into optics by Thomas Young. His Bakerian lecture on “The Theory of Light and Colours” (Phil. Trans., 1801) formulated the following hypotheses and propositions, and thereby laid the foundations of the wave theory:—

Hypotheses.

(i.) A luminiferous aether pervades the universe, rare and elastic in a high degree.

(ii.) Undulations are excited in this aether whenever a body becomes luminous.

(iii.) The sensation of different colours depends on the different frequency of vibrations excited by the light in the retina.

(iv.) All material bodies have an attraction for the aethereal medium, by means of which it is accumulated in their substance, and for a small distance around them, in a state of greater density but not of greater elasticity.

Propositions.

(i.) All impulses are propagated in a homogeneous elastic medium with an equable velocity.

(ii.) An undulation conceived to originate from the vibration of a single particle must expand through a homogeneous medium in a spherical form, but with different quantities of motion in different parts.

(iii.) A portion of a spherical undulation, admitted through an aperture into a quiescent medium, will proceed to be further propagated rectilinearly in concentric superfices, terminated laterally by weak and irregular portions of newly diverging undulations.

(iv.) When an undulation arrives at a surface which is the limit of mediums of different densities, a partial reflection takes place, proportionate in force to the difference of the densities.

(v.) When an undulation is transmitted through a surface terminating different mediums, it proceeds in such a direction that the sines of the angles of incidence and refraction are in the constant ratio of the velocity of propagation in the two mediums.

(vi.) When an undulation falls on the surface of a rarer medium, so obliquely that it cannot be regularly refracted, it is totally reflected at an angle equal to that of its incidence.

(vii.) If equidistant undulations be supposed to pass through a medium, of which the parts are susceptible of permanent vibrations somewhat slower than the undulations, their velocity will be somewhat lessened by this vibratory tendency; and, in the same medium, the more, as the undulations are more frequent.

(viii.) When two undulations, from different origins, coincide either perfectly or very nearly in direction, their joint effect is a combination of the motions belonging to each.

(ix.) Radiant light consists in undulations of the luminiferous aether.

In the Philosophical Transactions for 1802, Young refers to his discovery of “a simple and general law.” The law is that “wherever two portions of the same light arrive at the eye by different routes, either exactly or very nearly in the same direction, the light becomes most intense where the difference of the routes is a multiple of a certain length, and least intense in the intermediate state of the interfering portions; and this length is different for light of different colours.”

This appears to be the first use of the word interfering or interference as applied to light. When two portions of light by their co-operation cause darkness, there is certainly “interference” in the popular sense; but from a mechanical or mathematical point of view, the superposition contemplated in proposition viii. would more naturally be regarded as taking place without interference. Young applied his principle to the explanation of colours of striated surfaces (gratings), to the colours of thin plates, and to an experiment which we shall discuss later in the improved form given to it by Fresnel, where a screen is illuminated simultaneously by light proceeding from two similar sources. As a preliminary to these explanations we require an analytical expression for waves of simple type, and an examination of the effects of compounding them.

§ 2. Plane Waves of Simple Type.—Whatever may be the character of the medium and of its vibration, the analytical expression for an infinite train of plane waves is

A cos { | 2π | (Vt − x) + α }  
---|---|---  
λ  
  
(1),

in which λ represents the wave-length, and V the corresponding velocity of propagation. The coefficient A is called the amplitude, and its nature depends upon the medium and may here be left an open question. The phase of the wave at a given time and place is represented by α. The expression retains the same value whatever integral number of wave-lengths be added to or subtracted from x. It is also periodic with respect to t, and the period is

τ = λ/V

(2).

In experimenting upon sound we are able to determine independently τ, λ, and V; but on account of its smallness the periodic time of luminous vibrations eludes altogether our means of observation, and is only known indirectly from λ and V by means of (2).

There is nothing arbitrary in the use of a circular function to represent the waves. As a general rule this is the only kind of wave which can be propagated without a change of form; and, even in the exceptional cases where the velocity is independent of wave-length, no generality is really lost by this procedure, because in accordance with Fourier’s theorem any kind of periodic wave may be regarded as compounded of a series of such as (1), with wave-lengths in harmonical progression.

A well-known characteristic of waves of type (1) is that any number of trains of various amplitudes and phases, but of the same wave-length, are equivalent to a single train of the same type. Thus

ΣA cos { | 2π | (Vt − x) + α } = ΣA cos α·cos | 2π | (Vt − x) − ΣA sin α·sin | 2π | (Vt − x)  
---|---|---|---|---|---|---  
λ | λ | λ  
= P cos { | 2π | (Vt − x) + φ }  
---|---|---  
λ  
  
(3),

where

P2 = (ΣA cos α)2 = Σ(A sin α)2

(4),

tan φ = | Σ(A sin α)  
---|---  
Σ(A cos α)  
  
(5).

An important particular case is that of two component trains only.

A cos { | 2π | (Vt − x) + α } \+ A′ cos { | 2π | (Vt − x) + α′ }  
---|---|---|---|---  
λ | λ  
= P cos { | 2π | (Vt − x) + φ },  
---|---|---  
λ  
  
where

P2 = A2 \+ A′2 \+ 2AA′ cos (α − α′)

(6).

The composition of vibrations of the same period is precisely analogous, as was pointed out by Fresnel, to the composition of forces, or indeed of any other two-dimensional vector quantities. The magnitude of the force corresponds to the amplitude of the vibration, and the inclination of the force corresponds to the phase. A group of forces, of equal intensity, represented by lines drawn from the centre to the angular points of a regular polygon, constitute a system in equilibrium. Consequently, a system of vibrations of equal amplitude and of phases symmetrically distributed round the period has a zero resultant.

According to the phase-relation, determined by (α − α′), the amplitude of the resultant may vary from (A − A′) to (A + A′). If A′ and A are equal, the minimum resultant is zero, showing that two equal trains of waves may neutralize one another. This happens when the phases are opposite, or differ by half a (complete) period, and the effect is that described by Young as “interference.”

§ 3. Intensity.—The intensity of light of given wave-length must depend upon the amplitude, but the precise nature of the relation is not at once apparent. We are not able to appreciate by simple inspection the relative intensities of two unequal lights; and, when we say, for example, that one candle is twice as bright as another, we mean that two of the latter burning independently would give us the same light as one of the former. This may be regarded as the definition; and then experiment may be appealed to to prove that the intensity of light from a given source varies inversely as the square of the distance. But our conviction of the truth of the law is perhaps founded quite as much upon the idea that something not liable to loss is radiated outwards, and is distributed in succession over the surfaces of spheres concentric with the source, whose areas are as the squares of the radii. The something can only be energy; and thus we are led to regard the rate at which energy is propagated across a given area parallel to the waves as the measure of intensity; and this is proportional, not to the first power, but to the square of the amplitude.

§ 4. Resultant of a Large Number of Vibrations of Arbitrary Phase.—We have seen that the resultant of two vibrations of equal amplitude is wholly dependent upon their phase-relation, and it is of interest to inquire what we are to expect from the composition of a large number (n) of equal vibrations of amplitude unity, and of arbitrary phases. The intensity of the resultant will of course depend upon the precise manner in which the phases are distributed, and may vary from n2 to zero. But is there a definite intensity which becomes more and more probable as n is increased without limit?

The nature of the question here raised is well illustrated by the special case in which the possible phases are restricted to two opposite phases. We may then conveniently discard the idea of phase, and regard the amplitudes as at random positive or negative. If all the signs are the same, the intensity is n2; if, on the other hand, there are as many positive as negative, the result is zero. But, although the intensity may range from 0 to n2, the smaller values are much more probable than the greater.

The simplest part of the problem relates to what is called in the theory of probabilities the “expectation” of intensity, that is, the mean intensity to be expected after a great number of trials, in each of which the phases are taken at random. The chance that all the vibrations arc positive is 2−n, and thus the expectation of intensity corresponding to this contingency is 2−n·n2. In like manner the expectation corresponding to the number of positive vibrations being (n − 1) is

2−n·n·(n − 2)2,

and so on. The whole expectation of intensity is thus

1 | { 1·n2 \+ n·(n − 2)2 + | n(n − 1) | (n − 4)2  
---|---|---|---  
2n | 1·2  
+ | n (n − 1) (n − 2) | (n − 6)2 \+ ... }  
---|---|---  
1·2·3  
  
(1).

Now the sum of the (n + 1) terms of this series is simply n, as may be proved by comparison of coefficients of x2 in the equivalent forms

(ex \+ e−x)n = 2n (1 + ½ x2 \+ ... )n

= enx \+ ne(n−2)x + | n (n − 1) | e(n−4)x \+ ...  
---|---|---  
1·2  
  
The expectation of intensity is therefore n, and this whether n be great or small.

The same conclusion holds good when the phases are unrestricted. From (4), § 2, if A = 1,

P2 = n + 2Σ cos (α2 − α1)

(2),

where under the sign of summation are to be included the cosines of the ½ n(n − 1) differences of phase. When the phases are arbitrary, this sum is as likely to be positive as negative, and thus the mean value of P2 is n.

The reader must be on his guard here against a fallacy which has misled some high authorities. We have not proved that when n is large there is any tendency for a single combination to give the intensity equal to n, but the quite different proposition that in a large number of trials, in each of which the phases are rearranged arbitrarily, the mean intensity will tend more and more to the value n. It is true that even in a single combination there is no reason why any of the cosines in (2) should be positive rather than negative, and from this we may infer that when n is increased the sum of the terms tends to vanish in comparison with the number of terms. But, the number of terms being of the order n2, we can infer nothing as to the value of the sum of the series in comparison with n.

Indeed it is not true that the intensity in a single combination approximates to n, when n is large. It can be proved (Phil. Mag., 1880, 10, p. 73; 1899, 47. p. 246) that the probability of a resultant intermediate in amplitude between r and r + dr is

2 | e−r2/n r dr  
---|---  
n  
  
(3).

The probability of an amplitude less than r is thus

2 | ∫ r0 e−r2/n r dr = 1 − e−r2/n  
---|---  
n  
  
(4),

or, which is the same thing, the probability of an amplitude greater than r is

e−r2/n

(5).

The accompanying table gives the probabilities of intensities less than the fractions of n named in the first column. For example, the probability of intensity less than n is .6321.

.05 | .0488 |  .80 | .5506  
---|---|---|---  
.10 | .0952 | 1.00 | .6321  
.20 | .1813 | 1.50 | .7768  
.40 | .3296 | 2.00 | .8647  
.60 | .4512 | 3.00 | .9502  
  
It will be seen that, however great n may be, there is a fair chance of considerable relative fluctuations of intensity in consecutive combinations.

The mean intensity, expressed by

2 | ∫ ∞0 e−r2/n · r2 · r dr,  
---|---  
n  
  
is, as we have already seen, equal to n.

It is with this mean intensity only that we are concerned in ordinary photometry. A source of light, such as a candle or even a soda flame, may be regarded as composed of a very large number of luminous centres disposed throughout a very sensible space; and, even though it be true that the intensity at a particular point of a screen illuminated by it and at a particular moment of time is a matter of chance, further processes of averaging must be gone through before anything is arrived at of which our senses could ordinarily take cognizance. In the smallest interval of time during which the eye could be impressed, there would be opportunity for any number of rearrangements of phase, due either to motions of the particles or to irregularities in their modes of vibration. And even if we supposed that each luminous centre was fixed, and emitted perfectly regular vibrations, the manner of composition and consequent intensity would vary rapidly from point to point of the screen, and in ordinary cases the mean illumination over the smallest appreciable area would correspond to a thorough averaging of the phase-relationships. In this way the idea of the intensity of a luminous source, independently of any questions of phase, is seen to be justified, and we may properly say that two candles are twice as bright as one.  
  
---  
Fig. 1.  
  
§ 5. Interference Fringes.—In Fresnel’s fundamental experiment light from a point O (fig. 1) falls upon an isosceles prism of glass BCD, with the angle at C very little less than two right angles. The source of light may be a pin-hole through which sunlight enters a dark room, or, more conveniently, the image of the sun formed by a lens of short focus (1 or 2 in.). For actual experiment when, as usually happens, it is desirable to economize light, the point may be replaced by a line of light perpendicular to the plane of the diagram, obtained either from a linear source, such as the filament of an incandescent electric lamp, or by admitting light through a narrow vertical slit.

If homogeneous light be used, the light which passes through the prism will consist of two parts, diverging as if from points O1 and O2 symmetrically situated on opposite sides of the line CO. Suppose a sheet of paper to be placed at A with its plane perpendicular to the line OCA, and let us consider what illumination will be produced at different parts of this paper. As O1 and O2 are images of O, crests of waves must be supposed to start from them simultaneously. Hence they will arrive simultaneously at A, which is equidistant from them, and there they will reinforce one another. Thus there will be a bright band on the paper parallel to the edges of the prism. If P1 be chosen so that the difference between P1O2 and P1O1 is half a wave-length (i.e. half the distance between two successive crests), the two streams of light will constantly meet in such relative conditions as to destroy one another. Hence there will be a line of darkness on the paper, through P1, parallel to the edges of the prism. At P2, where O2P2 exceeds O1P2 by a whole wave-length, we have another bright band; and at P3, where O2P3 exceeds O1P3 by a wave-length and a half, another dark band; and so on. Hence, as everything is symmetrical about the bright band through A, the screen will be illuminated by a series of bright and dark bands, gradually shading into one another. If the paper screen be moved parallel to itself to or from the prism, the locus of all the successive positions of any one band will (by the nature of the curve) obviously be an hyperbola whose foci are O1 and O2. Thus the interval between any two bands will increase in a more rapid ratio than does the distance of the screen from the source of light. But the intensity of the bright bands diminishes rapidly as the screen moves farther off; so that, in order to measure their distance from A, it is better to substitute the eye (furnished with a convex lens) for the screen. If we thus measure the distance AP1 between A and the nearest bright band, measure also AO, and calculate (from the known material and form of the prism, and the distance CO) the distance O1O2, it is obvious that we can deduce from them the lengths of O1P2 and O2P2. Their difference is the length of a wave of the homogeneous light experimented with. Though this is not the method actually employed for the purpose (as it admits of little precision), it has been thus fully explained here because it shows in a very simple way the possibility of measuring a wave-length.

The difference between O1P1 and O2P1 becomes greater as AP1 is greater. Thus it is clear that the bands are more widely separated the longer the wave-length of the homogeneous light employed. Hence when we use white light, and thus have systems of bands of every visible wave-length superposed, the band A will be red at its edges, the next bright bands will be blue at their inner edges and red at their outer edges. But, after a few bands are passed, the bright bands due to one kind of light will gradually fill up the dark bands due to another; so that, while we may count hundreds of successive bright and dark bars when homogeneous light is used, with white light the bars become gradually less and less defined as they are farther from A, and finally merge into an almost uniform white illumination of the screen.

If D be the distance from O to A, and P be a point on the screen in the neighbourhood of A, then approximately

O1P − O2P = √{ D2 \+ (u + ½b)2 } − √{ D2 \+ (u − ½b)2 } = ub / D,

where O1O2 = b, AP = u.

Thus, if λ be the wave-length, the places where the phases are accordant are given by

u = nλD / b

(1),

n being an integer.

If the light were really homogeneous, the successive fringes would be similar to one another and unlimited in number; moreover there would be no place that could be picked out by inspection as the centre of the system. In practice λ varies, and (as we have seen) the only place of complete accordance for all kinds of light is at A, where u = 0. Theoretically, there is no place of complete discordance for all kinds of light, and consequently no complete blackness. In consequence, however, of the fact that the range of sensitiveness of the eye is limited to less than an “octave,” the centre of the first dark band (on either side) is sensibly black, even when white light is employed; but it should be carefully remarked that the existence of even one band is due to selection, and that the formation of several visible bands is favoured by the capability of the retina to make chromatic distinctions within the visible range.

The number of perceptible bands increases pari passu with the approach of the light to homogeneity. For this purpose there are two methods that may be used.

We may employ light, such as that from the soda flame, which possesses ab initio a rather high degree of homogeneity. If the range of wave-length included be 1⁄50000, a corresponding number of interference fringes may be made visible. The above was the number obtained by A. H. L. Fizeau. Using vacuum tubes containing, for example, mercury or cadmium vapour, A. A. Michelson has been able to go much farther. The narrowness of the bright line of light seen in the spectroscope, and the possibility of a large number of Fresnel’s bands, depend upon precisely the same conditions; the one is in truth as much an interference phenomenon as the other.

In the second method the original light may be highly composite, and homogeneity is brought about with the aid of a spectroscope. The analogy with the first method is closest if we use the spectroscope to give us a line of homogeneous light in simple substitution for the artificial flame. Or, following J. B. L. Foucault and Fizeau, we may allow the white light to pass, and subsequently analyse the mixture transmitted by a narrow slit in the screen upon which the interference bands are thrown. In the latter case we observe a channelled spectrum, with maxima of brightness corresponding to the wave-lengths bu/(nD). In either case the number of bands observable is limited solely by the resolving power of the spectroscope, and proves nothing with respect to the regularity, or otherwise, of the vibrations of the original light.

In lieu of the biprism, reflectors may be invoked to double the original source of light. In one arrangement two reflected images are employed, obtained from two reflecting surfaces nearly parallel and in the same plane. Glass, preferably blackened behind, may be used, provided the incidence be made sufficiently oblique. In another arrangement, due to H. Lloyd, interference takes place between light proceeding directly from the original source, and from one reflected image. Lloyd’s experiment deserves to be better known, as it may be performed with great facility and without special apparatus. Sunlight is admitted horizontally into a darkened room through a slit situated in a window-shutter, and, at a distance of 15 to 20 ft., is received at nearly grazing incidence upon a vertical slab of plate glass. The length of the slab in the direction of the light should not be less than 2 or 3 in., and for some special observations may advantageously be much increased. The bands are observed on a plane through the hinder vertical edge of the slab by means of a hand-magnifying glass of from 1 to 2 in. focus. The obliquity of the reflector is, of course, to be adjusted according to the fineness of the bands required.

From the manner of their formation it might appear that under no circumstances could more than half the system be visible. But according to Sir G. B. Airy’s principle (see below) the bands may be displaced if examined through a prism. In practice all that is necessary is to hold the magnifier somewhat excentrically. The bands may then be observed gradually to detach themselves from the mirror, until at last the complete system is seen, as in Fresnel’s form of the experiment.

The fringes now under discussion are those which arise from the superposition of two simple and equal trains of waves whose directions are not quite parallel. If the two directions of propagation are inclined on opposite sides of the axis of x at small angles α, the expressions for two components of equal amplitude are

cos | 2π | {Vt − x cos α − y sin α},  
---|---|---  
λ  
  
and

cos | 2π | {Vt − x cos α \+ y sin α},  
---|---|---  
λ  
  
so that the resultant is expressed by

2 cos | 2πy sin α | cos | 2π | {Vt − x cos α},  
---|---|---|---|---  
λ | λ  
  
from which it appears that the vibrations advance parallel to the axis of x, unchanged in type, and with a uniform velocity V/cos α. Considered as depending on y, the vibration is a maximum when y sin α is equal to O, λ, 2λ, 3λ, &c., corresponding to the centres of the bright bands, while for intermediate values ½λ, 3⁄2λ, &c., there is no vibration.

From (1) we see that the linear width Λ of the bands, reckoned from bright to bright or dark to dark, is

Λ = λD / b

(2).

The degree of homogeneity necessary for the approximate perfection of the nth Fresnel’s band may be found at once from (1) and (2). For if du be the change in u corresponding to the change dλ, then

du / Λ = ndλ / λ

(3).

Now clearly du must be a small fraction of Λ, so that dλ/λ must be many times smaller than 1/n, if the darkest places are to be sensibly black. But the phenomenon will be tolerably well marked if the proportional range of wave-length do not exceed 1/2n, provided, that is, that the distribution of illumination over this range be not concentrated towards the extreme parts.

So far we have supposed the sources at O1, O2 to be mathematically small. In practice, the source is an elongated slit, whose direction requires to be carefully adjusted to parallelism with the reflecting surface or surfaces. By this means an important advantage is gained in respect of brightness without loss of definition, as the various parts of the aperture give rise to coincident systems of bands.

The question of the admissible width of the slit requires consideration. We will suppose that the light issuing from various parts of the aperture is without permanent phase-relations, as when the slit is backed immediately by a flame, or by an incandescent filament. Regular interference can then only take place between light coming from corresponding parts of the two images, and a distinction must be drawn between the two ways in which the images may be situated relatively to one another. In Fresnel’s experiment, whether carried out with the mirrors or with the biprism, the corresponding parts of the images are on the same side; that is, the right of one corresponds to the right of the other, and the left of the one to the left of the other. On the other hand, in Lloyd’s arrangement the reflected image is reversed relatively to the original source; the two outer edges corresponding, as also the two inner. Thus in the first arrangement the bands due to various parts of the slit differ merely by a lateral shift, and the condition of distinctness is simply that the projection of the width of the slit be a small fraction of the width of the bands. From this it follows as a corollary that the limiting width is independent of the order of the bands under examination. It is otherwise in Lloyd’s method. In this case the centres of the systems of bands are the same, whatever part of the slit is supposed to be operative, and it is the distance apart of the images (b) that varies. The bands corresponding to the various parts of the slit are thus upon different scales, and the resulting confusion must increase with the order of the bands. From (1) the corresponding changes in u and b are given by

du = −nλD db/b2;

so that

du/Λ = −n db/b

(4).

If db represents twice the width of the slit, (4) gives a measure of the resulting confusion in the bands. The important point is that the slit must be made narrower as n increases if the bands are to retain the same degree of distinctness.

§ 6. Achromatic Interference Bands.—We have already seen that in the ordinary arrangement, where the source is of white light entering through a narrow slit, the heterogeneity of the light forbids the visibility of more than a few bands. The scale of the various band-systems is proportional to λ. But this condition of things, as we recognize from (2) (see § 5), depends upon the constancy of b, i.e. upon the supposition that the various kinds of light all come from the same place. Now there is no reason why such a limitation need be imposed. If we regard b as variable, we see that we have only to take b proportional to λ, in order to render the band-interval Λ independent of colour. In such a case the system of bands is achromatic, and the heterogeneity of the light is no obstacle to the formation of visible bands of high order.

These requirements are very easily met by the use of Lloyd’s mirrors, and of a diffraction grating (see Diffraction) with which to form a spectrum. White light enters the dark room through a slit in the window-shutter, and falls in succession upon a grating and an achromatic lens, so as to form a real diffraction spectrum, or rather a series of such, in the focal plane. The central image and all the lateral coloured images except one are intercepted by a screen. The spectrum which is allowed to pass is the proximate source of light in the interference experiment, and since the deviation of any colour from the central white image is proportional to λ, it is only necessary to arrange the mirror so that its plane passes through the white image in order to realize the conditions for the formation of achromatic bands.

When a suitable grating is at hand, the experiment in this form succeeds very well. If we are satisfied with a less perfect fulfilment of the achromatic conditions, the diffraction spectrum may be replaced by a prismatic one, so arranged that d(λ/b) = 0 for the most luminous rays. The bands are then achromatic in the sense that the ordinary telescope is so. In this case there is no objection to a merely virtual spectrum, and the experiment may be very simply executed with Lloyd’s mirror and a prism of (say) 20° held just in front of it.

The number of black and white bands shown by the prism is not so great as might be expected. The lack of contrast that soon supervenes can only be due to imperfect superposition of the various component systems. That the fact is so is at once proved by observing according to the method of Fizeau; for the spectrum from a slit at a very moderate distance out is seen to be traversed by bands. If the adjustment has been properly made, a certain region in the yellow-green is uninterrupted, while the closeness of the bands increases towards the other end of the spectrum. So far as regards the red and blue rays, the original bands may be considered to be already obliterated, but so far as regards the central rays, to be still fairly defined. Under these circumstances it is remarkable that so little colour should be apparent on direct inspection of the bands. It would seem that the eye is but little sensitive to colours thus presented, perhaps on account of its own want of achromatism.

§ 7. Airy’s Theory of the White Centre.—If a system of Fresnel’s bands be examined through a prism, the central white band undergoes an abnormal displacement, which has been supposed to be inconsistent with theory. The explanation has been shown by Airy (Phil. Mag., 1833, 2, p. 161) to depend upon the peculiar manner in which the white band is in general formed.

“Any one of the kinds of homogeneous light composing the incident heterogeneous light will produce a series of bright and dark bars, unlimited in number as far as the mixture of light from the two pencils extends, and undistinguishable in quality. The consideration, therefore, of homogeneous light will never enable us to determine which is the point that the eye immediately turns to as the centre of the fringes. What then is the physical circumstance that determines the centre of the fringes?

“The answer is very easy. For different colours the bars have different breadths. If then the bars of all colours coincide at one part of the mixture of light, they will not coincide at any other part; but at equal distances on both sides from that place of coincidence they will be equally far from a state of coincidence. If then we can find where the bars of all colours coincide, that point is the centre of the fringes.

“It appears then that the centre of the fringes is not necessarily the point where the two pencils of light have described equal paths, but is determined by considerations of a perfectly different kind.... The distinction is important in this and in other experiments.”

The effect in question depends upon the dispersive power of the prism. If v be the linear shifting due to the prism of the originally central band, v must be regarded as a function of λ. Measured from the original centre, the position of the nth bar is now

v + nλD / b.

The coincidence of the various bright bands occurs when this quantity is as independent as possible of λ, that is, when n is the nearest integer to

n = − | b |   | dv  
---|---|---|---  
D | dλ  
  
(1);

or, as Airy expresses it in terms of the width of a band (Λ), n = −dv/dΛ.

The apparent displacement of the white band is thus not v simply, but

v − Λdv / dΛ

(2).

The signs of dv and dΛ being opposite, the abnormal displacement is in addition to the normal effect of the prism. But, since dv/dΛ, or dv/dλ, is not constant, the achromatism of the white band is less perfect than when no prism is used.

If a grating were substituted for the prism, v would vary as Λ, and (2) would vanish, so that in all orders of spectra the white band would be seen undisplaced.

In optical experiments two trains of waves can interfere only when they have their origin in the same source. Otherwise, as it is usually put, there can be no permanent phase-relation, and therefore no regular interference. It should be understood, however, that this is only because trains of optical waves are never absolutely homogeneous. A really homogeneous train could maintain a permanent phase-relation with another such train, and, it may be added, would of necessity be polarized in its character. The peculiarities of polarized light with respect to interference are treated under Polarization of Light.

In a classical experiment interference-bands were employed to examine whether light moved faster or slower in glass than in air. For this purpose a very thin piece of glass may be interposed in the path of one of the interfering rays, and the resulting displacement of the bands is such as to indicate that the light passing through the glass is retarded. In a better form of the experiment two pieces of parallel glass cut from the same plate are interposed between the prism and the screen, so that the rays from O1 (fig. 1) pass through one part and those from O2 through the other. So long as these pieces are parallel, no shifting takes place, but if one be slightly turned, the bands are at once displaced. In the absence of dispersion the retardation R due to the plate would be independent of λ, and therefore completely compensated at the point determined by u = DR/b; but when there is dispersion it is accompanied by a fictitious displacement of the fringes on the principle explained by Airy, as was shown by Stokes.

Before quitting this subject it is proper to remark that Fresnel’s bands are more influenced by diffraction than their discoverer supposed. On this account the fringes are often unequally broad and undergo fluctuations of brightness. A more precise calculation has been given by H. F. Weber and by H. Struve, but the matter is too complicated to be further considered here. The observations of Struve appear to agree well with the corrected theory.

§ 8. Colours of Thin Plates.—These colours, familiarly known as those of the soap-bubble, are seen under a variety of conditions and were studied with some success by Robert Hooke under the name of “fantastical colours” (Micrographia, 1664). The inquiry was resumed by Sir Isaac Newton with his accustomed power (“Discourse on Light and Colours,” 1675, Opticks, book ii.), and by him most of the laws regulating these phenomena were discovered. Newton experimented especially with thin plates of air enclosed by slightly curved glasses, and the coloured rings so exhibited are usually called after him “Newton’s rings.”

The colours are manifested in the greatest purity when the reflecting surfaces are limited to those which bound the thin film. This is the case of the soap-bubble. When, as is in other respects more convenient, two glass plates enclosing a film of air are substituted, the light under examination is liable to be contaminated by that reflected from the outer surfaces. A remedy may be found in the use of wedge-shaped glasses so applied that the outer surfaces, though parallel to one another, are inclined to the inner operating surfaces. By suitable optical arrangements the two portions of light, desired and undesired, may then be separated.

In his first essay upon this subject Thomas Young was able to trace the formation of these colours as due to the interference of light reflected from the two surfaces of the plate; or, as it would be preferable to say, to the superposition of the two reflected vibrations giving resultants of variable magnitude according to the phase-relation. A difficulty here presents itself which might have proved insurmountable to a less acute inquirer. The luminous vibration reflected at the second surface travels a distance increased by twice the thickness of the plate, and it might naturally be supposed that the relative retardation would be measured by this quantity. If this were so, the two vibrations reflected from the surfaces of an infinitely thin plate would be in accordance, and the intensity of the resultant a maximum. The facts were notoriously the reverse. At the place of contact of Newton’s glasses, or at the thinnest part of a soap-film just before it bursts, the colour is black and not white as the explanation seems to require. Young saw that the reconciliation lies in the circumstance that the two reflections occur under different conditions, one, for example, as the light passes from air to water, and the second as it passes from water to air. According to mechanical principles the second reflection involves a change of sign, equivalent to a gain or loss of half an undulation. When a series of waves constituting any particular coloured light is reflected from an infinitely thin plate, the two partial reflections are in absolute discordance and, if of equal intensity, must give on superposition complete darkness. With the aid of this principle the sequence of colours in Newton’s rings is explained in much the same way as that of interference fringes (above, § 5).  
  
---  
Fig. 2.  
  
The complete theory of the colours of thin plates requires us to take account not merely of the two reflections already mentioned but of an infinite series of such reflections. This was first effected by S. D. Poisson for the case of retardations which are exact multiples of the half wave-length, and afterwards more generally by Sir G. B. Airy (Camb. Phil. Trans., 1832, 4, p. 409).

In fig. 2, ABF is the ray, perpendicular to the wave-front, reflected at the upper surface, ABCDE the ray transmitted at B, reflected at C and transmitted at D; and these are accompanied by other rays reflected internally 3, 5, &c., times. The first step is to calculate the retardation δ between the first and second waves, so far as it depends on the distances travelled in the plate (of index μ) and in air.

If the angle ABF = 2α, angle BCD = 2α′ and the thickness of plate = t, we have

δ = μ (BC + CD) − BG

= 2μBC − 2BC sin α sin α′ = 2μBC (l − sin2 α′)

= 2μt cos α′  
  
---  
  
(1).

In (1) α′ is the angle of refraction, and we see that, contrary to what might at first have been expected, the retardation is least when the obliquity is greatest, and reaches a maximum when the obliquity is zero or the incidence normal. If we represent all the vibrations by complex quantities, from which finally the imaginary parts are rejected, the retardation δ may be expressed by the introduction of the factor ε−iκδ, where i = √(−1), and κ = 2π/λ.

At each reflection or refraction the amplitude of the incident wave must be supposed to be altered by a certain factor which allows room for the reversal postulated by Young. When the light proceeds from the surrounding medium to the plate, the factor for reflection will be supposed to be b, and for refraction c; the corresponding quantities when the progress is from the plate to the surrounding medium will be denoted by e, f. Denoting the incident vibration by unity, we have then for the first component of the reflected wave b, for the second cefε−iκδ, for the third ce3fε−2iκδ, and so on. Adding these together, and summing the geometric series, we find

b + | cef ε−iκδ  
---|---  
1 − e2 ε−iκδ  
  
(2).

In like manner for the wave transmitted through the plate we get

cf  
---  
1 − e2 ε−iκδ  
  
(3).

The quantities b, c, e, f are not independent. The simplest way to find the relations between them is to trace the consequences of supposing δ = 0 in (2) and (3). This may be regarded as a development from Young’s point of view. A plate of vanishing thickness is ultimately no obstacle at all. In the nature of things a surface cannot reflect. Hence with a plate of vanishing thickness there must be a vanishing reflection and a total transmission, and accordingly

b + e = 0,  cf = l − e2

(4),

the first of which embodies Arago’s law of the equality of reflections, as well as the famous “loss of half an undulation.” Using these we find for the reflected vibration,

− | e(1 − ε−iκδ)  
---|---  
1 − e2 ε−iκδ  
  
(5),

and for the transmitted vibration

1 − e2  
---  
1 − e2 ε−iκδ  
  
(6).

The intensities of the reflected and transmitted lights are the squares of the moduli of these expressions. Thus

Intensity of reflected light = e2 | (1 − cos κδ)2 \+ sin2 κδ  
---|---  
(1 − e2 cos κδ)2 \+ e4 sin2 κδ  
= | 4e2 sin2 (½κδ)  
---|---  
1 − 2e2 cos κδ \+ e4  
  
(7);

Intensity of transmitted light = | (1 − e2)2  
---|---  
1 − 2e2 cos κδ \+ e4  
  
(8),

the sum of the two expressions being unity.

According to (7) not only does the reflected light vanish completely when δ = o, but also whenever ½κδ = nπ, n being an integer, that is, whenever δ = nλ. When the first and third mediums are the same, as we have here supposed, the central spot in the system of Newton’s ring is black, even though the original light contain a mixture of all wave-lengths. If the light reflected from a plate of any thickness be examined with a spectroscope of sufficient resolving power, the spectrum will be traversed by dark bands, of which the centre corresponds to those wave-lengths which the plate is incompetent to reflect. It is obvious that there is no limit to the fineness of the bands which may be thus impressed upon a spectrum, whatever may be the character of the original mixed light.  
  
---  
Fig. 3.  
  
The relations between the factors b, c, e, f have been proved, independently of the theory of thin plates, in a general manner by Stokes, who called to his aid the general mechanical principle of reversibility. If the motions constituting the reflected and refracted rays to which an incident ray gives rise be supposed to be reversed, they will reconstitute a reversed incident ray. This gives one relation; and another is obtained from the consideration that there is no ray in the second medium, such as would be generated by the operation alone of either the reversed reflected or refracted rays. Space does not allow of the reproduction of the argument at length, but a few words may perhaps give the reader an idea of how the conclusions are arrived at. The incident ray (IA) (fig. 3) being 1, the reflected (AR) and refracted (AF) rays are denoted by b and c. When b is reversed, it gives rise to a reflected ray b2 along AI, and a refracted ray bc along AG (say). When c is reversed, it gives rise to cf along AI, and ce along AG. Hence bc + ce = 0, b2 \+ cf = 1, which agree with (4). It is here assumed that there is no change of phase in the act of reflection or refraction, except such as can be represented by a change of sign.

When the third medium differs from the first, the theory of thin plates is more complicated, and need not here be discussed. One particular case, however, may be mentioned. When a thin transparent film is backed by a perfect reflector, no colours should be visible, all the light being ultimately reflected, whatever the wave-length may be. The experiment may be tried with a thin layer of gelatin on a polished silver plate. In other cases where a different result is observed, the inference is that either the metal does not reflect perfectly, or else that the material of which the film is composed is not sufficiently transparent. Some apparent exceptions to the above rule, exhibited by thin films of collodion resting upon silver surfaces, have been described by R. W. Wood (Physical Optics, p. 143), who attributes the very curious effects observed to frilling of the collodion film.

For study of the colours of thin plates there are no more interesting subjects than the soap-film. For projection the films may be stretched across vertical rings of iron wire coated with paraffin. In their undisturbed condition they thin from the top, and the colours are disposed in horizontal bands. If, as suggested by Brewster, a jet of wind issuing from a small nozzle and supplied from a well-regulated bellows be allowed to impinge obliquely, parts of the film are set in rotation, and displays of colours may be exhibited to a large audience, astonishing by their brilliance and by the rapidity with which they change. Permanent films, analogous to soap-films, are best obtained by Glew’s method. A few drops of celluloid varnish are poured upon the surface of water contained in a large dish. After evaporation of the solvent, the films may be picked up upon rings of iron wire.

As a variant upon Newton’s rings, interesting effects may be obtained by the partial etching of the surfaces of picked pieces of plate-glass. A surface is coated in parallel stripes with paraffin wax and treated with dilute hydrofluoric acid for such a time (found by preliminary trials) as is required to eat away the exposed portions to a depth of one quarter of the mean wave-length of light. Two such prepared surfaces pressed in the crossed position into suitable contact exhibit a chess-board pattern. Where two uncorroded, or where two corroded, parts overlap, the colours are nearly the same; but where a corroded and an uncorroded surface meet, a strongly contrasted colour is developed. The combination lends itself to projection and the pattern seen upon the screen is very beautiful if proper precautions are taken to eliminate the white light reflected from the first and fourth surfaces of the plates (see Nature, 1901, 64, 385).

Theory and observation alike show that the transmitted colours of a thin plate, e.g. a soap film or a layer of air, are very inferior to those reflected. Specimens of ancient glass, which have undergone superficial decomposition, on the other hand, sometimes show transmitted colours of remarkable brilliancy. The probable explanation, suggested by Brewster, is that we have here to deal not merely with one, but with a series of thin plates of not very different thicknesses. It is evident that with such a series the transmitted colours would be much purer, and the reflected much brighter, than usual. If the thicknesses are strictly equal, certain wave-lengths must still be absolutely missing in the reflected light; while on the other hand a constancy of the interval between the plates will in general lead to a special preponderance of light of some other wave-length for which all the component parts as they ultimately emerge are in agreement as to phase.

On the same principle are doubtless to be explained the colours of fiery opals, and, more remarkable still, the iridescence of certain crystals of potassium chlorate. Stokes showed that the reflected light is often in a high degree monochromatic, and that it is connected with the existence of twin planes. A closer discussion appears to show that the twin planes must be repeated in a periodic manner (Phil. Mag., 1888, 26, 241, 256; also see R. W. Wood, Phil. Mag., 1906).

A beautiful example of a similar effect is presented by G. Lippmann’s coloured photographs. In this case the periodic structure is actually the product of the action of light. The plate is exposed to stationary waves, resulting from the incidence of light upon a reflecting surface (see Photography).

All that can be expected from a physical theory is the determination of the composition of the light reflected from or transmitted by a thin plate in terms of the composition of the incident light. The further question of the chromatic character of the mixtures thus obtained belongs rather to physiological optics, and cannot be answered without a complete knowledge of the chromatic relations of the spectral colours themselves. Experiments upon this subject have been made by various observers, and especially by J. Clerk Maxwell (Phil. Trans., 1860), who has exhibited his results on a colour diagram as used by Newton. A calculation of the colours of thin plates, based upon Maxwell’s data, and accompanied by a drawing showing the curve representative of the entire series up to the fifth order, has been given by Rayleigh (Edin. Trans., 1887). The colours of Newton’s scale are met with also in the light transmitted by a somewhat thin plate of doubly-refracting material, such as mica, the plane of analysis being perpendicular to that of primitive polarization.

The same series of colours occur also in other optical experiments, e.g. at the centre of the illuminated area when light issuing from a point passes through a small round aperture in an otherwise opaque screen.

The colours of which we have been speaking are those formed at nearly perpendicular incidence, so that the retardation (reckoned as a distance), viz. 2μt cos α′, as sensibly independent of λ. This state of things may be greatly departed from when the thin plate is rarer than its surroundings, and the incidence is such that α′ is nearly equal to 90°, for then, in consequence of the powerful dispersion, cos α′ may vary greatly as we pass from one colour to another. Under these circumstances the series of colours entirely alters its character, and the bands (corresponding to a graduated thickness) may even lose their coloration, becoming sensibly black and white through many alternations (Newton’s Opticks, bk. ii.; Fox-Talbot, Phil. Mag., 1836, 9, p. 40l). The general explanation of this remarkable phenomenon was suggested by Newton.

Let us suppose that plane waves of white light travelling in glass are incident at angle α upon a plate of air, which is bounded again on the other side by glass. If μ be the index of the glass, α′ the angle of refraction, then sin α′ = μ sin α; and the retardation, expressed by the equivalent distance in air, is

2t sec α′ − μ·2t tan α′ sin α = 2t cos α′;

and the retardation in phase is 2t cos α′/λ, λ being as usual the wave-length in air.

The first thing to be noticed is that, when α approaches the critical angle, cosα′ becomes as small as we please, and that consequently the retardation corresponding to a given thickness is very much less than at perpendicular incidence. Hence the glass surfaces need not be so close as usual.

A second feature is the increased brilliancy of the light. According to (7) the intensity of the reflected light when at a maximum (sin ½κγ = 1) is 4e2/(1 + e2)2. At perpendicular incidence e is about 1⁄5, and the intensity is somewhat small; but, as cos α′ approaches zero, e approaches unity, and the brilliancy is much increased.

But the peculiarity which most demands attention is the lessened influence of a variation in λ upon the phase-retardation. A diminution of λ of itself increases the retardation of phase, but, since waves of shorter wave-length are more refrangible, this effect may be more or less perfectly compensated by the greater obliquity, and consequent diminution in the value of cos α′. We will investigate the conditions under which the retardation of phase is stationary in spite of a variation of λ.

In order that λ−1 cos α′ may be stationary, we must have

λ sin α′ dα′ \+ cos α′ dλ = 0,

where (α being constant)

cos α′ dα′ = sin α dμ.

Thus

cot2 α′ = | λ |   | dμ  
---|---|---|---  
μ | dλ  
  
(9),

giving α′ when the relation between μ and λ is known.

According to A. L. Cauchy’s formula, which represents the facts very well throughout most of the visible spectrum,

μ = A + Bλ−2

(10),

so that

cot2 α′ = | 2B | = | 2(μ − A)  
---|---|---|---  
λ2μ | μ  
  
(11).

If we take, as for Chance’s “extra-dense flint,” B = .984 × 10-10, and as for the soda lines, μ = 1.65, λ = 5.89 × 10-6, we get

α′ = 79°30′.

At this angle of refraction, and with this kind of glass, the retardation of phase is accordingly nearly independent of wave-length, and therefore the bands formed, as the thickness varies, are approximately achromatic. Perfect achromatism would be possible only under a law of dispersion

μ2 = A′ − B′λ2.

If the source of light be distant and very small, the black bands are wonderfully fine and numerous. The experiment is best made (after Newton) with a right-angled prism, whose hypothenusal surface may be brought into approximate contact with a plate of black glass. The bands should be observed with a convex lens, of about 8 in. focus. If the eye be at twice this distance from the prism, and the lens be held midway between, the advantages are combined of a large field and of maximum distinctness.

If Newton’s rings are examined through a prism, some very remarkable phenomena are exhibited, described in his twenty-fourth observation (Opticks; see also Place, Pogg. Ann., 1861, 114, 504). “When the two object-glasses are laid upon one another, so as to make the rings of the colours appear, though with my naked eye I could not discern above eight or nine of those rings, yet by viewing them through a prism I could see a far greater multitude, insomuch that I could number more than forty.... And I believe that the experiment may be improved to the discovery of far greater numbers.... But it was on but one side of these rings, namely, that towards which the refraction was made, which by the refraction was rendered distinct, and the other side became more confused than when viewed with the naked eye....

“I have sometimes so laid one object-glass upon the other that to the naked eye they have all over seemed uniformly white, without the least appearance of any of the coloured rings; and yet by viewing them through a prism great multitudes of those rings have discovered themselves.”

Newton was evidently much struck with these “so odd circumstances”; and he explains the occurrence of the rings at unusual thicknesses as due to the dispersing power of the prism. The blue system being more refracted than the red, it is possible under certain conditions that the nth blue ring may be so much displaced relatively to the corresponding red ring as at one part of the circumference to compensate for the different diameters. A white stripe may thus be formed in a situation where without the prism the mixture of colours would be complete, so far as could be judged by the eye.

The simplest case that can be considered is when the “thin plate” is bounded by plane surfaces inclined to one another at a small angle. By drawing back the prism (whose edge is parallel to the intersection of the above-mentioned planes) it will always be possible so to adjust the effective dispersing power as to bring the nth bars to coincidence for any two assigned colours, and therefore approximately for the entire spectrum. The formation of the achromatic band, or rather central black band, depends indeed upon the same principles as the fictitious shifting of the centre of a system of Fresnel’s bands when viewed through a prism.

But neither Newton nor, as would appear, any of his successors has explained why the bands should be more numerous than usual, and under certain conditions sensibly achromatic for a large number of alternations. It is evident that, in the particular case of the wedge-shaped plate above specified, such a result would not occur. The width of the bands for any colour would be proportional to λ, as well after the displacement by the prism as before; and the succession of colours formed in white light and the number of perceptible bands would be much as usual.

The peculiarity to be explained appears to depend upon the curvature of the surfaces bounding the plate. For simplicity suppose that the lower surface is plane (y = 0), and that the approximate equation of the upper surface is y = a + bx2, a being thus the least distance between the plates. The black of the nth order for wave-length λ occurs when

½nλ = a + bx2

(12);

and thus the width (δx) at this place of the band is given by

½λ = 2bxδx

(13);

or

δx = | λ | = | λ  
---|---|---|---  
4bx | 4√b · √(½nλ − a)  
  
(14).

If the glasses be in contact, as is usually supposed in the theory of Newton’s rings, a = 0, and δx∞λ1/2, or the width of the band of the nth order varies as the square root of the wave-length, instead of as the first power. Even in this case the overlapping and subsequent obliteration of the bands is greatly retarded by the use of the prism, but the full development of the phenomenon requires that α should be finite. Let us inquire what is the condition in order that the width of the band of the nth order may be stationary, as λ varies. By (14) it is necessary that the variation of λ2/(½nλ − a) should vanish. Hence a = ¼nλ, so that the interval between the surfaces at the place where the nth band is formed should be half due to curvature and half to imperfect contact at the place of closest approach. If this condition be satisfied, the achromatism of the nth band, effected by the prism, carries with it the achromatism of a large number of neighbouring bands, and thus gives rise to the remarkable effects described by Newton. Further developments are given by Lord Rayleigh in a paper “On Achromatic Interference Bands” (Phil. Mag., 1889, 28, pp. 77, 189); see also E. Mascart, Traité d’optique.

In Newton’s rings the variable element is the thickness of the plate, to which the retardation is directly proportional, and in the ideal case the angle of incidence is constant. To observe them the eye is focused upon the thin plate itself, and if the plate is very thin no particular precautions are necessary. As the plate thickens and the order of interference increases, there is more and more demand for homogeneity in the light, and we may have recourse to a sodium-flame or a helium vacuum tube. At the same time the disturbing influence of obliquity increases. Unless the aperture of the eye is reduced, the rays reaching it from even the same point of the plate are differently affected, and complications ensue tending to impair the distinctness of the bands. To obviate this disturbance it is best to work at incidences as nearly as possible perpendicular.  
  
---  
Fig. 4.  
  
The bands seen when light from a soda flame falls upon nearly parallel surfaces are often employed as a test of flatness. Two flat surfaces can be made to fit, and then the bands are few and broad, if not entirely absent; and, however the surfaces may be presented to one another, the bands should be straight, parallel and equidistant. If this condition be violated, one or other of the surfaces deviates from flatness. In fig. 4, A and B represent the glasses to be tested, and C is a lens of 2 or 3 ft. focal length. Rays diverging from a soda flame at E are rendered parallel by the lens, and after reflection from the surfaces are recombined by the lens at E. To make an observation, the coincidence of the radiant point and its image must be somewhat disturbed, the one being displaced to a position a little beyond, and the other to a position a little in front of the diagram. The eye, protected from the flame by a suitable screen, is placed at the image, and being focused upon AB, sees the field traversed by bands. The reflector D is introduced as a matter of convenience to make the line of vision horizontal.

These bands may be photographed. The lens of the camera takes the place of the eye, and should be as close to the flame as possible. With suitable plates, sensitized by cyanin, the exposure required may vary from ten minutes to an hour. To get the best results, the hinder surface of A should be blackened, and the front surface of B should be thrown out of action by the superposition of a wedge-shaped plate of glass, the intervening space being filled with oil of turpentine or other fluid having nearly the same refraction as glass. Moreover, the light should be purified from blue rays by a trough containing solution of bichromate of potash. With these precautions the dark parts of the bands are very black, and the exposure may be prolonged much beyond what would otherwise be admissible.

By this method it is easy to compare one flat with another, and thus, if the first be known to be free from error, to determine the errors of the second. But how are we to obtain and verify a standard? The plan usually followed is to bring three surfaces into comparison. The fact that two surfaces can be made to fit another in all azimuths proves that they are spherical and of equal curvatures, but one convex and the other concave, the case of perfect flatness not being excluded. If A and B fit one another, and also A and C, it follows that B and C must be similar. Hence, if B and C also fit one another, all three surfaces must be flat. By an extension of this process the errors of three surfaces which are not flat can be found from a consideration of the interference bands which they present when combined in three pairs.

The free surface of undisturbed water is almost ideally flat, and, as Lord Rayleigh (Nature, 1893, 48, 212) has shown, there is no great difficulty in using it as a standard of comparison. Following the same idea we may construct a parallel plate by superposing a layer of water upon mercury. If desired, the superior reflecting power of the mercury may be compensated by the addition of colouring matter to the water.

Haidinger’s Rings dependent on Obliquity.—It is remarkable that the well-known theoretical investigation, undertaken with the view of explaining Newton’s rings, applies more directly to a different system of rings discovered at a later date.

The results embodied in equations (1) to (8) have application in the first instance to plates whose surfaces are absolutely parallel, though doubtless they may be employed with fair accuracy when the thickness varies but slowly.

We have now to consider t constant and α′ variable in (1). If α′ be small,

δ = 2μt (1 − ½α′2) = 2μt − tα2 / μ

(15);

and since the differences of δ are proportional to α2, the law of formation is the same as for Newton’s rings, where α′ is constant and t proportional to the square of the distance from the point of contact. In order to see these rings distinctly the eye must be focused, not upon the plate, but for infinitely distant objects.

The earliest observation of rings dependent upon obliquity appears to have been made by W. von Haidinger (Pogg. Ann., 1849, 77, p. 219; 1855, 96, p. 453), who employed sodium light reflected from a plate of mica (e.g. 0.2 mm. thick). The transmitted rays are the easier to see in their completeness, though they are necessarily somewhat faint. For this purpose it is sufficient to look through the mica, held close to the eye and perpendicular to the line of vision, at a sheet of white paper or card illuminated by a sodium flame. Although Haidinger omitted to consider the double refraction of the mica and gave formulae not quite correct for even singly refracting plates, he fully appreciated the distinctive character of the rings, contrasting Berührungsringe und Plattenringe. The latter may appropriately be named after him. Their tardy discovery may be attributed to the technical difficulty of obtaining sufficiently parallel plates, unless it be by the use of mica or by the device of pouring water upon mercury. Haidinger’s rings were rediscovered by O. R. Lummer (Wied. Ann., 1884, 23, p. 49), who pointed out the advantages they offer in the examination of plates intended to be parallel.

The illumination depends upon the intensity of the monochromatic source of light, and upon the reflecting power of the surfaces. If R be the intensity of the reflected light we have from (7)

1 | = 1 + | (1 − e2)2 | ;  
---|---|---|---  
R | 4e2 sin2 (½κδ)  
  
from which we see that if e = 1 absolutely, 1/R = R = 1 for all values of δ. If e = 1 very nearly, R = 1 nearly for all values of δ for which sin2 (½κδ) is not very small. In the light reflected from an extended source, the ground will be of full brightness corresponding to the source, but it will be traversed by narrow dark lines. By transmitted light the ground, corresponding to general values of the obliquity, will be dark, but will be interrupted by narrow bright rings, whose position is determined by sin ½(κδ) = 0. In permitting for certain directions a complete transmission in spite of a high reflecting power (e) of the surfaces, the plate acts the part of a resonator.

There is no transparent material for which, unless at high obliquity, e approaches unity. In C. Fabry and A. Pérot’s apparatus the reflections at nearly perpendicular incidence are enhanced by lightly silvering the surfaces. In this way the advantage of narrowing the bright rings is attained in great measure without too heavy a sacrifice of light. The plate in the optical sense is one of air, and is bounded by plates of glass whose inner silvered surfaces are accurately flat and parallel. The outer surfaces need only ordinary flatness, and it is best that they be not quite parallel to the inner ones. The arrangement constitutes a spectroscope, inasmuch as it allows the structure of a complex spectrum line to be directly observed. If, for example, we look at a sodium flame, we see in general two distinct systems of narrow bright circles corresponding to the two D-lines. With particular values of the thickness of the plate of air the two systems may coincide so as to be seen as a single system, but a slight alteration of thickness will cause a separation.

It will be seen that in this apparatus the optical parts are themselves of extreme simplicity; but they require accuracy of construction and adjustment, and the demand in these respects is the more severe the further the ideal is pursued of narrowing the rings by increase of reflecting power. Two forms of mounting are employed. In one instrument, called the interferometer, the distance between the surfaces—the thickness of the plate—is adjustable over a wide range. In its complete development this instrument is elaborate and costly. The actual measurements of wave-lengths by Fabry and Pérot were for the most part effected by another form of instrument called an étalon or interference-gauge. The thickness of the optical plate is here fixed; the glasses are held up to metal knobs, acting as distance-pieces, by adjustable springs, and the final adjustment to parallelism is effected by regulating the pressure exerted by these springs. The distance between the surfaces may be 5 or 10 mm.

The theory of the comparison of wave-lengths by means of this apparatus is very simple, and it may be well to give it, following closely the statement of Fabry and Pérot (Ann. chim. phys., 1902, 25, p. 110). Consider first the cadmium radiation λ treated as a standard. It gives a system of rings. Let P be the ordinal number of one of these rings, for example the first counting from the centre. This integer is supposed known. The order of interference at the centre will be p = P + ε. We have to determine this number ε, lying ordinarily between 0 and 1. The diameter of the ring under consideration increases with ε; so that a measure of the diameter allows us to determine the latter. Let t be the thickness of the plate of air. The order of interference at the centre is p = 2t/λ. This corresponds to normal passage. At an obliquity i the order of interference is p cos i. Thus if x be the angular diameter of the ring P, p cos ½x = P; or since x is small,

p = P (1 + 1⁄8x2).

In like manner, from observations upon another radiation λ′ to be compared with λ, we have

p′ = P′ (1 + 1⁄8x′2);

whence if t be treated as an absolute constant,

λ′ | = | P | ( 1 + | x2 | − | x′2 | )  
---|---|---|---|---|---|---|---  
λ |  P′ | 8 | 8  
  
(16).

The ratio λ/λ′ is thus determined as a function of the angular diameters x, x′ and of the integers P, P′. If P, say for the cadmium red line, is known, an approximate value of λ/λ′ will usually suffice to determine what integral value must be assigned to P′, and thence by (16) to allow of the calculation of the corrected ratio λ′/λ.

In order to find P we may employ a modified form of (16), viz.,

P′ | = | λ | ( 1 + | x2 | − | x′2 | )  
---|---|---|---|---|---|---|---  
P | λ′ | 8 | 8  
  
(17),

using spectrum lines, such as the cadmium red and the cadmium green, for which the relative wave-lengths are already known with accuracy from A. A. Michelson’s work. To test a proposed integral value of P (cadmium red), we calculate P′ (cadmium green) from (17), using the observed values of x, x′. If the result deviates from an integer by more than a small amount (depending upon the accuracy of the observations), the proposed value of P is to be rejected. In this way by a process of exclusion the true value is ultimately arrived at (Rayleigh, Phil. Mag., 1906, 685). It appears that by Fabry and Pérot’s method comparisons of wave-lengths may be made accurate to about one-millionth part; but it is necessary to take account of the circumstance that the effective thickness t of the plate is not exactly the same for various wave-lengths as assumed in (16).

§ 9. Newton’s Diffusion Rings.—In the fourth part of the second book of his Opticks Newton investigates another series of rings, usually (though not very appropriately) known as the colours of thick plates. The fundamental experiment is as follows. At the centre of curvature of a concave looking-glass, quicksilvered behind, is placed an opaque card, perforated by a small hole through which sunlight is admitted. The main body of the light returns through the aperture; but a series of concentric rings are seen upon the card, the formation of which was proved by Newton to require the co-operation of the two surfaces of the mirror. Thus the diameters of the rings depend upon the thickness of the glass, and none are formed when the glass is replaced by a metallic speculum. The brilliancy of the rings depends upon imperfect polish of the anterior surface of the glass, and may be augmented by a coat of diluted milk, a device used by Michel Ferdinand, duc de Chaulnes. The rings may also be well observed without a screen in the manner recommended by Stokes. For this purpose all that is required is to place a small flame at the centre of curvature of the prepared glass, so as to coincide with its image. The rings are then seen surrounding the flame and occupying a definite position in space.

The explanation of the rings, suggested by Young, and developed by Herschel, refers them to interference between one portion of light scattered or diffracted by a particle of dust, and then regularly refracted and reflected, and another portion first regularly refracted and reflected and then diffracted at emergence by the same particle. It has been shown by Stokes (Camb. Trans., 1851, 9, p. 147) that no regular interference is to be expected between portions of light diffracted by different particles of dust.

In the memoir of Stokes will be found a very complete discussion of the whole subject, and to this the reader must be referred who desires a fuller knowledge. Our limits will not allow us to do more than touch upon one or two points. The condition of fixity of the rings when observed in air, and of distinctness when a screen is used, is that the systems due to all parts of the diffusing surface should coincide; and it is fulfilled only when, as in Newton’s experiments, the source and screen are in the plane passing through the centre of curvature of the glass.  
  
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Fig. 5.  
  
As the simplest for actual calculation, we will consider a little further the case where the glass is plane and parallel, of thickness t and index μ, and is supplemented by a lens at whose focus the source of light is placed. This lens acts both as collimator and as object-glass, so that the combination of lens and plane mirror replaces the concave mirror of Newton’s experiment. The retardation is calculated in the same way as for thin plates. In fig. 5 the diffracting particle is situated at B, and we have to find the relative retardation of the two rays which emerge finally at inclination θ, the one diffracted at emergence following the path ABDBIE, and the other diffracted at entrance and following the path ABFGH. The retardation of the former from B to I is 2μt + BI, and of the latter from B to the equivalent place G is 2μBF. Now FB = t sec θ′, θ′ being the angle of refraction; BI = 2t tan θ′sin θ; so that the relative retardation F is given by

R = 2μt {1 + μ−1 tan θ′ sin θ − sec θ′) = 2μt (1 − cos θ′).

If θ, θ′ be small, we may take

R = 2tθ2 / μ

(1).

as sufficiently approximate.

The condition of distinctness is here satisfied, since R is the same for every ray emergent parallel to a given one. The rays of one parallel system are collected by the lens to a focus at a definite point in the neighbourhood of the original source.

The formula (1) was discussed by Herschel, and shown to agree with Newton’s measures. The law of formation of the rings follows immediately from the expression for the retardation, the radius of the ring of nth order being proportional to n and to the square root of the wave-length.

§ 10. Interferometer.—In many cases it is necessary that the two rays ultimately brought to interference should be sufficiently separated over a part of their course to undergo a different treatment; for example, it may be desired to pass them through different gases.  
  
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Fig. 6.  
Fig. 7.  
  
A simple modification of Young’s original experiment suffices to solve this problem. Light proceeding from a slit at A (fig. 6) perpendicular to the plane of the paper, falls upon a collimating lens B whose aperture is limited by two parallel and rather narrow slits of equal width. The parallel rays CE, DF (shown broken in the figure) transmitted by these slits are brought to a focus at G by the lens EF where they form an image of the original slit A. This image is examined with an eye-piece of high magnifying power. The interference bands at G undergo displacement if the rays CE, DF are subjected to a relative retardation. Consider what happens at the point G, which is the geometrical image of A. If all is symmetrical so that the paths CE, DF are equal, there is brightness. But if, for example, CE be subjected to a relative retardation of half a wave-length, the brightness is replaced by darkness, and the bands are shifted through half a band-interval.

An apparatus of this kind has been found suitable for determining the refractivity of gases, especially of gases available only in small quantities (Proc. Roy. Soc., 1896, 59, p. 198; 1898, 64, p. 95). There is great advantage in replacing the ordinary eye-piece by a simple cylindrical magnifier formed of a glass rod 4 mm. in diameter. Under these conditions a paraffin lamp sufficed to illuminate the slit at A, and allowed the refractivities of gases to be compared to about one-thousandth part.

If the object be to merely see the bands in full development the lenses of the above apparatus may be dispensed with. A metal or pasteboard tube 10 in. long carries at one end a single slit (analogous to A) and at the other a double slit (analogous to C, D). This double slit, which requires to be very fine, may be made by scraping two parallel lines with a knife on a piece of silvered glass. The tube is pointed to a bright light, and the eye, held close behind the double slit, is focused upon the far slit.

§ 11. Other Refractometers.—In another form of refractometer, employed by J. C. Jamin, the separations are effected by reflections at the surfaces of thick plates. Two thick glass mirrors, exactly the same in all respects, are arranged as in fig. 7. The first of the two interfering rays is that which is reflected at the first surface of the first reflector and at the second surface of the second reflector. The second ray undergoes reflection at the second surface of the first reflector and at the first surface of the second reflector. Upon the supposition that the plates are parallel and equally thick, the paths pursued by these two rays are equal. P represents a thin plate of glass interposed in the path of one ray, by which the bands are shifted.  
  
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Fig. 8.  
  
In Jamin’s apparatus the two rays which produce interference are separated by a distance proportional to the thickness of the mirrors, and since there is a practical limit to this thickness, it is not possible to separate the two rays very far. In A. A. Michelson’s interferometer there is no such restriction. “The light starts from source S (fig. 8) and separates at the rear of plate A, part of it being reflected to the plane mirror C, returning exactly, on its path through A, to O, where it may be observed by a telescope or received upon a screen. The other part of the ray goes through the glass plate A, passes through B, and is reflected by the plane mirror D, returns on its path to the starting point A, where it is reflected so as nearly to coincide with the first ray. The plane parallel glass B is introduced to compensate for the extra thickness of glass which the first ray has traversed in passing twice through the plate A. Without it the two paths would not be optically identical, because the first would contain more glass than the second. Some light is reflected from the front surface of the plate A, but its effect may be rendered insignificant by covering the rear surface of A with a coating of silver of such thickness that about equal portions of the incident light are reflected and transmitted. The plane parallel plates A and B are worked originally in one piece, which is afterwards cut in two. The two pieces are placed parallel to one another, thus ensuring exact equality in the two optical paths AC and AD” (see Michelson, Light-Waves and their Uses, Chicago, 1903).

The adjustments of this apparatus are very delicate. Of the fully silvered mirrors C, D, the latter must be accurately parallel to the image of the former. For many purposes one of the mirrors, C, must be capable of movement parallel to itself, usually requiring the use of very truly constructed ways. An escape from this difficulty may be found in the employment of a layer of mercury, standing on copper, the surface of which automatically assumes the horizontal position.

Michelson’s apparatus, employed to view an extended field of homogeneous light, exhibits Haidinger’s rings, and if all is in good order the dark parts are sensibly black. As the order of interference increases, greater and greater demand is made upon the homogeneity of the light. Thus, if the illumination be from a sodium flame, the rings are at first distinct, but as the difference of path increases the duplicity of the bright sodium line begins to produce complications. After 500 rings, the bright parts of one system coincide with the dark parts of the other (Fizeau), and if the two systems were equally bright all trace of rings would disappear. A little later the rings would again manifest themselves and, after 1000 had gone by, would be nearly or quite as distinct as at first. And these alternations of distinctness and indistinctness would persist until the point was reached at which even a single sodium line was insufficiently homogeneous. Conversely, the changes of visibility of the rings as the difference of path increases give evidence as to the duplicity of the line. In this way Michelson obtained important information as to the constitution of the approximately homogeneous lines obtained from electrical discharge through attenuated metallic vapours. Especially valuable is the vacuum tube containing cadmium. The red line proved itself to be single and narrow in a high degree, and the green line was not far behind.

But although in Michelson’s hands the apparatus has done excellent spectroscopic work, it is not without its weak points. A good deal of labour is required to interpret the visibility curves, and in some cases the indications are actually ambiguous. For instance, it is usually impossible to tell on which side of the principal component a feebler companion lies. It would seem that for spectroscopic purposes this apparatus must yield to that of Fabry and Pérot, in which multiple reflections are utilized; this is a spectroscope in the literal sense, inasmuch as the constitution of a spectrum line is seen by simple inspection.

(R.)

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1 The word “interference” as formed, on the false analogy of such words as “difference,” from “to interfere,” which originally was applied to a horse striking (Lat. ferire) one foot or leg against the other.

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