Chapter VI: Marine Insurance (3)

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 ([alpha] - [alpha]'),
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."

S 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.

S 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 n^2 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 n^2; 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 n^2, 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).n^2. In like
manner the expectation corresponding to the number of positive
vibrations being (n - 1) is

2^(-n).n.(n -)^2,

and so on. The whole expectation of intensity is thus

1 / n(n - 1)
----- ( 1.n^2 + n.(n - 2)^2 + -------- (n - 4)^2
2^(n) \ 1.2

n(n - 1)(n - 2) \
+ ---------------(n - 6)^2 + ... ) (1).
1.2.3 /

Now the sum of the (n + 1) terms of this series is simply n, as may be
proved by comparison of coefficients of x^2 in the equivalent forms

(e^x + e^(-x))^n = 2^n(1 + (1/2)x^2 + ... )^n

n(n - 1)
= e^(nx) + ne^[(n-2)x] + -------- 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), S 2, if A = 1,

P^2 = n + 2[Sigma] cos ([alpha]2 - [alpha]1) (2),

where under the sign of summation are to be included the cosines of
the (1/2)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 P^2 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 n^2, 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^(-r^2/n) rdr (3),
n

The probability of an amplitude less than r is thus

_
2 /r
-- | e^(-r^2/n) rdr = 1 - e^(-r^2/n) (4),
n _/0

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

e^(-r^2/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 / [oo]
-- | e^(-r^2/n).r^2.rdr,
n _/ 0

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.

S 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 = \/ {D^2 + (u + 1/2 b)^2} - \/{D^2 + (u - 1/2 b)^2} = ub/D,

where O1O2 = b, AP = u.

Thus, if [lambda] be the wave-length, the places where the phases are
accordant are given by

u = n[lambda]D/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 [lambda] 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 [alpha],
the expressions for two components of equal amplitude are

2[pi]
cos -------- {Vt - x cos [alpha] - y sin [alpha]},
[lambda]

and

2[pi]
cos -------- {Vt - x cos [alpha] + y sin [alpha]},
[lambda]

so that the resultant is expressed by

2[pi]y sin [alpha] 2[pi]
2 cos ------------------ cos -------- {Vt - x cos [alpha]},
[lambda] [lambda]

from which it appears that the vibrations advance parallel to the axis
of x, unchanged in type, and with a uniform velocity V/cos [alpha].
Considered as depending on y, the vibration is a maximum when y sin
[alpha] is equal to O, [lambda], 2[lambda], 3[lambda], &c.,
corresponding to the centres of the bright bands, while for
intermediate values (1/2)[lambda], (3/2)[lambda], &c., there is no
vibration.

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

[Lambda] = [lambda]D/b (2).

The degree of homogeneity necessary for the approximate perfection of
the n^(th) Fresnel's band may be found at once from (1) and (2). For
if du be the change in u corresponding to the change d[lambda], then

du/[Lambda] = nd[lambda]/[lambda] (3).

Now clearly du must be a small fraction of [Lambda], so that
d[lambda]/[lambda] 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 = -n[lambda]D db/b^2;

so that

du/[Lambda] = -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.

S 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 [lambda]. But this condition of things, as we recognize from (2) (see S 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 [lambda], in order to render the band-interval [Lambda] 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 [lambda], 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([lambda]/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 deg. 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.

S 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 [lambda]. Measured from the
original centre, the position of the n^(th) bar is now

v + n[lambda]D/b.

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

b dv
n = - --- --------- (1);
D d[lambda]

or, as Airy expresses it in terms of the width of a band ([Lambda]), n
= -dv/d[Lambda].

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

v - [Lambda]dv/d[Lambda] (2).

The signs of dv and d[Lambda] being opposite, the abnormal
displacement is in addition to the normal effect of the prism. But,
since dv/d[Lambda], or dv/d[lambda], 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 [Lambda],
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 [lambda], 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.

S 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, S 5).

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 [delta] between the first and second waves, so far as
it depends on the distances travelled in the plate (of index [mu]) and
in air.

If the angle ABF = 2[alpha], angle BCD = 2[alpha]' and the thickness
of plate = t, we have

[delta] = [mu](BC + CD) - BG
= 2[mu]BC - 2BC sin [alpha] sin [alpha]' = 2[mu]BC(l - sin^2 [alpha]')
= 2[mu]t cos [alpha]' (1).

In (1) [alpha]' 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 [delta] may be expressed by the
introduction of the factor [epsilon]^(-i[kappa][delta]), where i =
[root](-1), and [kappa] = 2[pi]/[lambda].

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
cef[Epsilon]^(-i[kappa][delta]), for the third
ce^3f[epsilon]^{-2i[kappa delta]}, and so on. Adding these together,
and summing the geometric series, we find

cef[epsilon]^(-i[kappa][delta])
b + ----------------------------------- (2).
1 - e^2[epsilon]^(-i[kappa][delta])

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

cf
----------------------------------- (3).
1 - e^2[epsilon]^(-i[kappa][delta])

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 [delta] = 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 - e^2 (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 - [epsilon]^(-i[kappa][delta])}
----------------------------------- (5),
1 - e^2[epsilon]^(-i[kappa][delta])

and for the transmitted vibration

1 - e^2
----------------------------------- (6).
1 - e^2[epsilon]^(-i[kappa][delta])

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

Intensity of reflected light =

(1 - cos [kappa][delta])^2 + sin^2 [kappa][delta]
= e^2 --------------------------------------------------------
(1 - e^2cos [kappa][delta])^2 + e^4 sin^2 [kappa][delta]

4e^2sin^2 {(1/2)[kappa][delta]}
= -------------------------------- (7);
1 - 2e^2 cos[kappa][delta] + e^4

(1 - e^2)^2
Intensity of transmitted light = -------------------------------- (8),
1 - 2e^2cos [kappa][delta] + e^4

the sum of the two expressions being unity.

According to (7) not only does the reflected light vanish completely
when [delta] = 0, but also whenever (1/2)[kappa][delta] = n[pi], n
being an integer, that is, whenever [delta] = n[lambda]. 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.

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 b^2 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, b^2 + 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. 2[mu]t cos [alpha]', as sensibly independent of [lambda].
This state of things may be greatly departed from when the thin plate
is rarer than its surroundings, and the incidence is such that
[alpha]' is nearly equal to 90 deg., for then, in consequence of the
powerful dispersion, cos [alpha]' 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 [alpha] upon a plate of air, which is bounded again
on the other side by glass. If [mu] be the index of the _glass_,
[alpha]' the angle of refraction, then sin [alpha]' = [mu] sin
[alpha]; and the retardation, expressed by the equivalent distance in
air, is

2t sec [alpha]' - [mu].2t tan [alpha]' sin [alpha] = 2t cos [alpha]';

and the retardation in _phase_ is 2t cos [alpha]'/[lambda], [lambda]
being as usual the wave-length in air.

The first thing to be noticed is that, when [alpha] approaches the
critical angle, cos[alpha]' 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
(1/2)[kappa][gamma] = 1) is 4e^2/(1 + e^2)^2. At perpendicular
incidence e is about 1/5, and the intensity is somewhat small; but, as
cos[alpha]' 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 [lambda] upon the phase-retardation. A
diminution of [lambda] 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 [alpha]'. We
will investigate the conditions under which the retardation of phase
is stationary in spite of a variation of [lambda].

In order that [lambda]^(-1) cos [alpha]' may be stationary, we must have

[lambda] sin [alpha]' d[alpha]' + cos [alpha]' d[lambda] = 0,

where ([alpha] being constant)

cos [alpha]' d[alpha]' = sin [alpha] d[mu].

[lambda] d[mu]
Thus cot^2 [alpha]' = - -------- --------- (9),
[mu] d[lambda]

giving [alpha]' when the relation between [mu] and [lambda] is known.

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

[mu] = A + B[lambda]^(-2) (10),

so that

2B 2([mu] - A)
cot^2 [alpha]' = -------------- = ----------- (11).
[lambda]^2[mu] [mu]

If we take, as for Chance's "extra-dense flint," B = .984 X 10^(-10),
and as for the soda lines, [mu] = 1.65, [lambda] = 5.89 X 10^(-6), we
get

[alpha]' = 79 deg. 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

[mu]^2 = A' - B'[lambda]^2.

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 n^(th) 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 n^(th) 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 [lambda],
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 = [alpha] + bx^2, a being thus the least
distance between the plates. The black of the n^(th) order for
wave-length [lambda] occurs when

(1/2)n[lambda] = [alpha] + bx^2 (12);

and thus the width ([delta]x) at this place of the band is given by

(1/2)[lambda] = 2bx[delta]x (13);

[lambda] [lambda]
or [delta]x = -------- = ----------------------------------------- (14).
4bx 4[root]b.[root]((1/2)n[lambda] - [alpha])