_
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.
[Illustration: FIG. 1.]
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].