F. (François) Arago.

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destroy each other, that darkness may result from the
superposition of two portions of light. But when this

Since, if we take the partial difierentials in i-espect to i and to x,

du , du ^ ,

-^ = w cos int — kx) -T- =^ k cos int — kx)

dt dx

— = — vP'U ■ — ^^= — li^u

dt^ dx-2

Whence, c?% n''^ d^u

dl^ " ~ W dx^
And since that wave-function goes through all its changes while t,
increases to — and the velocity f = -r- the time of the undulation





_27r








1




In


T


n


and


V





T





•Zn



Whence, n = and A; = —

Or the formula becomes (adopting an arbitrary coefficient, a, for
the amplitude of vibration which is wholly independent of the other
quantities)

277

M = a sin -r- {vt — x).

A

n
Here it is to be observed, all depends on the coefScient-r being

constant. To obtain a similar equation with a variable velocity or
refraction is the object of the researches of M. Cauchy.

The more extended views of M. Cauchy have led to the deduction
of analogous, but more complex, equations, exhibiting resulting ex-
pressions for the displacement, in three rectangular directions; besides
including in the analysis a coefficient which expresses the variable
relation of the velocity which gives the theoretical explanation of un-
equal refrangibility. These forms thus include the deduction of
transverse vibrations, as a direct consequence of the first assump-
tions, as to the constitution of an sBthereal medium. But, with refer-
ence to light, considered as homogeneous, the conditions admit of
great simplification; which is best shown in that form of the investi-
gation which was pursued by Sir J. Lubbock (Pliibs. Mag. Nov.
1837), where, if the fourth powers of the disturbed distances of the
molecules are neglected, the equations are at once reduced to the
form above.

The object of M. Cauchy's researches here alluded to was to ex-



THEORY OF INTERFERENCE. 215

property of the rays has been once established, is it not
still more exti-aordinary that we can deprive them of it?

plain the unequal refrangibility of light. To give some general notion
of the nature of the subject, we may here briefly observe, that in the
explanation of refraction before given, [Life of Malus, note,] it is
clear that as the inclination of the common tangent to the contempo-
nincous circular waves determines the refraction, this depends on the
diminution of the wave length within the denser medium; and if this
inclination be deteniiined for a ray of any given wave length, then for
another whose wave length within the medium is different, and in a
given ratio to the former, the radii of the contemporaneous waves will
be in the same ratio as the former, or their difference from the former
will be in the same ratio, consequentlj' the common tangent of these
second circles will not be parallel to the first, but inclined at a differ-
ent angle: or the angle of refraction will be different. Thus if for
any particular [primary ra}^ the wave length within the medium be

X
Til ^ that of the incident ray being /I and u the index for that
i«y -^ '^

rav, /I = 6 sin i.

1
then 2,1 = h sin r = ~

or in other words, the refraction will be different from each primary
ray. But //, and a, do. not follow any simple ratio. The more com-
plex expression on which that relation depends, is the result of M.
Cauchy's theory, viz: —

[See Professor Powell's Treatise on the Undulatory Theory, sect.
vi.]

Experimentally, the transverse vibrations receive their main sup-
port from the analysis of the coloured tints, developed in polarized
light by the interposition of plates of crystal (such as those of mica,
selenite, &c.), when examined by an analj'zer.

Young ascribed these colours generally to interference ; but both
Fresnel and Arago pointed out that this explanation was incomplete.
Why did it only take place in polarized light, and even then not until
the analyzer had been applied? These questions could not be an-
swered till another law had been discovered; as it soon after was, by
the joint labours of those two philosophers.

It was clear that in polarization all the vibrations were performed
in one and the same plane, in whatever direction they might be exe-



216



tliat a given ray loses it momentarily, and that another
given ray, on the contrary, is deprived of it for ever ?
The theory of interferences, considered in this point of
view, seems more like the reveries of a disordered brain,
than the exact, inevitable consequence of numberless
experiments, clear of all possible objection. And fur-

cuted. But it was not until after lengthened investigation that the
two philosophers just named succeeded in establishing experimen-
tally the important law (obvious as it now seems,) that ^^ polarized rays
can only interfere iclien they are polarized in the same plane.'''' If they
were polai'ized in rectangular planes (for example), no interference
could result, were all other conditions ever so perfectly fulfilled. Now,
this could only be explained on the supposition of the vibrations
being performed in planes transverse to the ray. Granting that in a
ray polarized in one plane all the vibrations take place in one plane,
(whether in the same plane or perpendicular to it,) it is then readily
seen that when the vibrations of two raj's are at right angles to each
other, there can be no mutual destruction, or mutual cooperation. It
is only when they are in the same plane that this can occur.

This principle was at length found to supply the explanation of the
polarized tints. Every ray of the light (p) originally polarized iu one




plane, in traversing the crystal plate (c) was divided into two; an
ordinary (o) and an extraordinary (e); all those of the one kind, o, o^,
o", &c. being polarized in one plane, and all of the other, e, c', e'', &c.,
in a plane at right angles to the last. But in each ray o, and e, di-
verge from each other by a very small angle. The whole pencil also
diverges at a small angle from p; thus, the only rays which can coin-
cide in direction, will be a ray o, of one set, with a ray e' of the next;
— (/, with e", &c. &c., and as these are unequallj' retarded in differ-
ent degrees according to their inclination, they would be in a con-
dition to give interference, were it not that being polarized in places



POLARIZATION. 217

ther, it is not only on account of its singularity that this
theory ought to command the attention of the physicist ;
Fresnel found it the key to all the beautiful phenomena
of colours, which are produced in plates of crystal pos-
sessing double refraction ; he analyzed them in all their
details ; he determined their most hidden laws ; he
proved that they were only particular cases of inter-
ferences. He thus overturned from their base many
scientific romances to which these phenomena had given
birth, and which had secured more than one pros-
elyte, whether by their striking nature or the distin-
guished merit of their authors. In a word, here, as in
every branch of science which is advancing towards per-
fection, the facts have seemed complicated only because
we examined them at too near a distance and with too
microscopic a view ; but at the same time, by a more
enlarged conception, their causes have been found to be
more simple than we might have expected.

rOLARIZATION.

Although I am aware at what point we risk tiring even
the most kindly disposed audience when we speak long

at rif/lil angles to each other they could not. It only required then
the action of the analyzer (a) to resolve each vibration again into



two, at right angles, of which two sets in a plane perpendicular to
that of analyzation are suppressed; and two in that plane transmit-
ted; and which., consequently, being in parallel planes, are able to
give interference, and produce the observed coloured tints.

8KC. SEK. 10



218 FRESNEL.

on the same subject, I find myself still carried back, by
the nature of Fresnel's labours, to the subject of double
refraction: but, this time, instead cf occupying myself
with the manner in which the rays divide in passing
throu'i^h certain crystals, I will examine the permanent
modifications which they receive : I will present, in one
word, the principal features of the new branch of optics
which bears the name of polarization ofUrjht.

Every ray of light falling even perpendicularly on any
surface, natural or artificial, of the transparent crystals of
carbonate of lime, called also calc spar, or Iceland spar,
is divided into two. One portion passes tlirough the
crystal without deviation, which we call the ordinary I'ay ;
the other undergoes a sensible refraction, and for that
reason has very justly the name of the extraordinary ray.
Both the ordinary and extraordinary i-ay lie in one plane
perpendicular to the face of the crystal. The considera-
tion of this plane is important, for it is this which deter-
mines the direction Avhich the extraordinary ray will
take ; and in consequence a special name has been given
to it, " the principal section."

These points being premised, I will suppose, to fix the
ideas, that a particular crystal of calc spar has its princi-
pal section directed north and south. Below this, and at
any distance, we will place another similar crystal turned
similarly ; that is, so th.at its principal section shall also
lie in the meridian. What will result from this disposi-
tion, if light traverse the whole system ? A single ray
impinges on the first crystal but it emerges in two rays :
each of these again seems as if it should undergo a double
refraction in the second crystal ; and thence we might
expect four emergent rays. Yet this does not happen.
The rays emanating from the first crystal are not divided



DOUBLE REFRACTION. 219

again by the second. The ordinary ray remains an ordi-
nary ray, and the extraordinary undergoes solely an
extraordinary refraction. Thus, in traversing the first
crystal, the luminous rays have changed their nature ;
they have lost one of their former characteristics, that of
constantly undergoing double refraction in traversing Ice-
land crystal.*

It is necessary that we should fully bear in mind what
rays of light are, and then, perhaps, we shall admit that
an experiment, by the aid of which they change their
original properties in so manifest a way, deserves to be
known even by«those to whom science is merely an object
of curiosity.

The idea which in the first instance presents itself to
the mind, when we wish to explain this singular result of
which I have just given an account, consists in supposing
that in every ray there might exist two distinct species of
molecules : that the one species must always undergo the
ordinary refraction ; the other, the extraordinary alone.
But a very simple experiment upsets this hypothesis en-
tirely. In fact, when the principal section of the second
crystal, instead of being directed north and south as above
supposed, is pointed east and west, the ray which was the
ordinary ray in the first crystal, becomes the extraordi-
nary in the second ; and reciprocally.

What, then, is there different in reality between the
two experiments which give results so dissimilar ? There
is one circumstance, very simple, and full of import at
first sight : it is, that at first the principal section of the
second crystal cuts the rays coming from the first through
their north and south sides, and in the second case, through
their east and west sides.

* For illustration of this subject, see note to the Life of Malus.



220 FKESNEL.

There must be then, in each of these rays, north and
south sides in some way different from their east and
west sides. And further, tlie north and south sides of
the ordinary ray ouglit to have precisely the same prop-
erties as the east and west sides of the extraordinary ray ;
so that if this hist ray make a quarter of a circuit about
tlie hne of its length it will be impossible to distinguish
one from the other. The rays of light are so subtle that
thousands of millions of these rays can pass simultane-
ously through the eye of a needle without interfering :
yet we find ourselves obliged to take into account the
idea of their sides, and to recognize, on their opposite
sides, dissimila7' properties.

When speaking of a magnet, natural or artificial phys-
icists affirm it to have poles. They mean only that cer-
tain points on its surface are found endowed with certain
properties which are not found, or at least only show
themselves feebly, at any other points. We have, then,
equal reason to say the same thing of the ordinary and
extraordinary rays of light which proceed from the divis-
ion of the beam which passes through Iceland spar ; and
in contradistinction to the natural rays in which all points
appear alike, we may rightly call them polarized rays.

In order, however, that we may not extend beyond its
proper limits the analogy of a polarized ray and a mag-
net, it is important to remark well that in the ray, the
poles diametrically opposite appear to possess exactly the
same properties ; whilst the dissimilar poles are situated
on sides of the ray whose positions are at right angles to
each other.

The lines resembling diameters, which join the similar
poles, in every ray deserve particular attention. When-
ever, in two distinct rays, these lines are parallel, we say



DOUBLE REFRACTION. 221

that the rays are polarized in the same plane. There is,
consequently, no need to add that two rays polarized at
right angles to each other must have their similar poles
in two directions perpendicular the one to the other.

The two rays, the ordinary and the extraordinary for
example, given by any ci-ystal are always polarized at
right angles to each other.

All that I have just said of polarization of light was
recognized by Huyghens and Newton before the end of
the 17th century; and never, certainly, had a more curi-
ous subject for research been offered to the meditations
of experimenters. Nevertheless, we must pass over an
interval of a century after that pei-iod before we find, I
do not say any fresh discoveries, but even any more
reseai'ches for the object of carrying out this branch of
optics.

The history of all sciences presents a multitude of sin-
gular incidents of a similar kind. In the progress of
each science there occur periodically certain epochs when,
after great efforts, men usually suppose themselves to
have arrived at a limit in their advance. Then expei'i-
raenters are in general timid ; they fancy themselves
chargeable with a want of modesty, with a sort of profa-
nation, if they dare to lay an indiscreet hand on the bar-
riers which their illustrious predecessors have erected ;
and thus they generally content themselves with perfect-
ing the numerical elements, or filling up some deficien-
cies, bestowing on the inquiry a labour often arduous,
and which yet scarcely attracts any notice from the
world.

In a word, the experiments of Huyghens had clearly
established the fact that double refraction modifies the
original properties of light in such a manner that, after



222 FRESNEL.

having once undergone this modification, the rays remain
single, or again subdivide into two, according to the direc-
tion in which they fall upon a second crystal presented to
them. But do these modifications show a relation exclu-
sively to double refraction ? do all their other properties
remain uninfluenced ?

It was from the labours of one of our most distinguished
colleagues (like Fresnel, early snatched away from the
sciences of which he was the hope) that we have been
enabled to answer these important questions. Malus dis-
covered, in fact, that, in the act of reflexion, polarized
rays are differently affected from common rays : the lat-
ter, as every one knows, are partially reflected when they
fall even on transparent bodies, whatever may be the
angle of incidence, and whatever the position of the re-
flecting surface with respect to the sides of the ray.
When, on the contrary, the case is one of polarized light,
there is always one situation of the reflecting surface,
relatively to the poles, or sides, in which all reflexion
disappears if in this situation the reflexion take at a
particular incidence, which is different for each reflecting
surface, according to the nature of the substance of which
it is foi'med.

If, after this curious observation, double refraction
ceased to be the 07ili/ means of distinguishing polarized
from common light, at least it seemed to be the only way
by which rays of light could become polarized. But
soon a new experiment of Malus taught the scientific
world, to its great surprise, that there existed other
methods, far less abstruse, for producing this modifica-
tion. The most simple phenomenon of optics, the re-
flexion of light from a transparent mirror, is a powerful
means of producing polarization. Light, which is re-



PLANE POLARIZATION. 223

fleeted at the surface of water at an angle of 37°, or
from the surface of glass at an inclination of 35° 25', is
as completely polarized as the two rays, ordinary and
extraordinary, proceeding from a crystal of Iceland spar.

The reflexion of light long ago occupied observers in
the age of Plato and of Euclid : since that epoch it had
been the object of thousands of experiments, of hundreds
of theoretical speculations ; the law according to which it
proceeds serves as the basis of a great number of instru-
ments, ancient and modern. Among the multitude of
enlightened minds, of men of genius, of skilful artists,
who, during more than 2300 years, have been occupied
with this phenomenon, no one ever aimed at any other
object than the means of making tlie rays divide, or of
causing them to diverge or converge ; no one ever imag-
ined that reflected light ought not to possess all the same
properties as the incident light, or that a change of path
would be the cause of a change of nature. Generations
of observers thus succeeded each other during several
thousands of years, every day toucliing closely on the
most beautiful discoveries without actually making them.

Malus, as I have already explained, gave a means of
polarizing light different from that which Huyghens had
formerly announced. But the polarizations produced by
the two methods Avere identically the same. The re-
flected rays and those which proceed out of an Iceland
crystal possess exactly the same properties. Since that
time a member of this Academy (Arago) has discovered
a kind of polarization* entirely distinct, and which mani-

* It maj' be necessary for some readers to explain that, in this some-
■what paradoxical mode of speaking, the autlior is referring to his own
discoverj' of the polarized tints; and his meaning is simply that if, in
polarized light, there be placed a thin film, e. g., of selenite or mica,
and it be viewed through a doubly refracting crystal as an analj^zer,



224 FRESNEL.

fests itself in a different way from that of difference of
intensity. The rays subjected to it, for example, always
give two images in traversing calc spar ; but these images
are each entirely tinted with a bright and uniform colour.
Thus, though the incident light may be white, the ordi-
nary ray may be entirely red, orange, yellow, green, blue,
or violet, according to the direction in which the prin-
cipal section of the crystal cuts the ray : and as to the
extraordinary ray, it will not suffice to say that it never
resembles the ordinary ; we must say that it differs from
it as widely as possible ; that if the one, for example, is
coloured red, the other shows a bright green, and so on
for the rest of the prismatic tints.

"When this new kind of polarized rays are reflected
from a transparent mirror, we perceive otiier phenomena
not less curious. Let us conceive, in fact, to fix the ideas,
that one of these rays be vertical, and that it fall on a re-
flector of pure glass at an angle of about 35°, this mirror
may be on the right side of the ray : and the inclination
remaining constant, it may be turned to its left, before it,
or behind it, or in any intermediate position. We may
remember that the incident ray was white ; then, in any
of these positions of the glass reflector, the ray will not
have this colour : it will be now red, now orange, yellow,
green, blue, indigo, violet, according to the side on which
the glass presents itself to the incident ray ; it is, in fact,
precisely in this order that the tints succeed one another,

both the images will be coloured, and their tints complementary. The
originally polarized light is divided again into two oppositely polarized
pencils in passing through the film, or as Professor J. Forbes has termed
it, Dipohirized; others had termed it \>Kpolarized. This is what Arago
here calls a new and entirely distinct kind of polarization; though the
term is, perhaps, not very happily applied. This is what was explained
at large in a previous note.



CHROMATIC POLARIZATION. 225

as we gradually make the mirror go through all possible
changes of position, Here there ai'e not only four poles
placed in two rectangular directions, which we must ad-
mit in the constitution of the ray, but we see that there
are thousands ; that every point in the circumference
round the ray has a special character ; that every face
which it presents produces in the reflexion a particular
tint. This strange dislocation of the natural ray (I may
be allowed this word, sjnce it exactly expresses the fact)
thus affords the means of decomjoosing white light by
means of reflexion. The colours, it must be avowed,
have not all the homogeneity of those which Newton ob-
tained by the prism ; but also the object from which they
originate does not undergo any distortion, as in prismatic
refraction : and in a multitude of researches this is a point
of material importance.

To discover whether a ray has received the polariza-
tion of Huyghens and Malus, or that of which I have
just spoken, and which wc call chromatic polarization, it
suffices as we have seen, to make it undergo double re-
fraction : but from the fact that a ray in traversing a
crystal of Iceland spar always gives two images of white
light and of equal intensity, it will not follow that it is
formed originally of common light : this is again the
discovery of Fresnel.* It is he who first pointed out

* The author would have expressed his meaning more clearly to
general apprehension if he had said, that natural or unpolarized white
light, on traversing Iceland spar, gives two white images in all posi-
tions: an ordinarily polarized ray does not; but there is a kind of
light which gives always two images, and j^et is not unpolarized : this
is the circularly polarized light, discovered by Fresnel. One test
which distinguishes it from common light is, that on interposing a
crystallized plate of seleuite, mica, &c., before receiving the light on
10*



226 FRESNEL.

that a ray may have the same properties round all points
of its circumference, and yet not be common light. To
show by a single example that these two species of light
comport themselves differently, and ought not to be con-
founded, I Avill observe that, in undergoing double re-
fraction, a natural ray after traversing a plate of crystal
gives two white images, wliile under the same conditions
the ray of Fresnel is decomposed into two beams, each
hrilliantly coloured.

This new modification, which, having no reference to
the different sides of a ray, has been designated circidar
polay-ization, can be impressed upon rays ordinarily
polarized, by making them undergo two successive total
reflexions from the internal surfaces of a piece of glass
suitably formed.* The pleasure of having his name as-
sociated with a new kind of polarization hitherto unsus-
pected, would probably have sufficed for the vanity of
an ordinary experimenter, and his researches would not
have extended beyond that point. But Fresnel was
actuated by more elevated sentiments ; in his eyes
nothing seemed to have been done while any thing re-
mained to do. He sought, therefore, if there were not

the double refracting crystal, the two images in the former case will
be always white, in the later coloured.

* In the instance mentioned, Fresnel showed, by a remarkable in-
stance of theoretical prediction, that a ray polarized at 45° to the
plane of incidence, and twice reflected internally from glass, will
emerge in the condition of two ra\-s polarized in planes at right



Online LibraryF. (François) AragoBiographies of distinguished scientific men (Volume 2) → online text (page 17 of 38)