F. (François) Arago.

Biographies of distinguished scientific men (Volume 2) online

. (page 20 of 38)
Online LibraryF. (François) AragoBiographies of distinguished scientific men (Volume 2) → online text (page 20 of 38)
Font size
QR-code for this ebook


projectiles), the fact, and above all the laws, of inter-
In the refracted ray the intensities of the residuary portions respec-
tively will be

1 {l — h'-2) ini
i il + k'-2) in K.
Here the second is always the greater: and the refracted ray con-
tains an excess of light polarized perpendicularly to the plane of inci-
dence. The difference or quantity of light polarized is the same as in
the reflected ray. Hence the light will be completely polarized at
any incidence for which either of the expressions (3.) or (5.) vanishes.
No value of i will make (5.) vanish, since we can never have i = r.
But the expression (3.) becomes = when i-l-9'=90°. In this case
the light is comjAetely polarized in the plane of incidence. But in this
case we have also

sin i . . ,1
cos I = sin r = or tan i = H;

which is Brewster's law ; also if i + r "^90" we have — tan (i + r).

Also at this incidence h. the incident light is reflected, wholly polar-
ized in I; 5 is also transmitted wholly polarized in K. This is the case
referred to by Arago in the text. From (5.) also another remarkable
inference follows : if the reflexion be internal, or the ray be incident
on the second surface of a dense medium, we have r greater than i,
or

sin (J — r)

that is, t\\Q phase of the reflected vibration is changed by 180° equiva-

•1
lent to a difference of — in route, from what it would be in reflexion

at the first surface at the same incidence. This explains the sujjposed
assunqyiion of the half undulation in Newton's rings.

Again: if a polarized ray be incident on a reflecting surface with
its plane of vibrations inclined to the plane of incidence (i), at an
angle (a), its vibration (A) may be resolved into two, one in the plane
(i), and one perpendicular to it (k), in the ratio of sin a and cos a,
or after reflexion we shall have for the respective amplitudes (5.)
and (3.)

kl sin a, and h> cos a.
These by composition will give a resultant ray polarized in a plane
(p), inclined to (i) by angle {/3), and we have from the formulas (5.)
and (3.)

, , . cos (J + r)

tan S =■ — tan a - .

cos (i — /•)



254 FRESNEL.

ference appear wholly inexplicable. I will add besides,
that none of the partisans of the system of emission

This formula exhibits remarkable changes at successive incidences:
at incidences less than that of complete polarization, the new plane of
polarization (as indicated by the sign of the tangent) deviates on the
side of the plane opposite to that of polarization (p) — at (i,) inci-
dences c/reater, it deviates on the same side as p; results which ar/ree
exactly with numerous and accurate observations of Fresnel, Arago, and
Brewster.

We have also the following results of this last formula:

While a has any finite value, when i = 0, /J = a, or the plane of
polarization is unchanged.

When (»■ + ?•) = 90°, /:? = 0, or at the angle of complete polarization
p coincides with I.

When i — 90°, jS = a again, or p has its original position.

If a = 0, hi sin a = 0, and if at the same time ( i + r) — 90°, then
let = 0, or we also see that at the polarizing angle an incident ray polar-
ized in I loill cease to give any reflected ray ; which agrees with the
observation originally made by Malus.

From the same formulas another more curious inference was made
by Fresnel as follows : In passing out of a denser into a rarer medium,

. . 1

in general it is well known if t = 90°, sm t = — .

Consequently a ray making this incidence internally on the bound-
ing surface will not be refracted out; and at incidences more oblique
is experimentally found to be totally reflected internally : theoretically,
the conversation of vis viva would require that the whole vibratory
force, since none of it is expended on refraction, must be occupied in
communicating vibrations internally, which can only produce internal
waves or internal total reflexion.

Now at the critical incidence, in the formulas for h' and kt, sin
(i — r) = cos i, sin (i + r) = sin i and tan (i — r) = cot i tan {i + r)
= tan ii; whence hi = I and k = 1, which accords with total re-
flexion.

At incidences greater than this the values become imaginary; and
by introducing into them empirically certain terms i multiplied by
v^^ir Fresnel obtained in such cases an expression of the form,

27r

(cos e -\- V— 1 sin 0) sin - (vt — x)

A

1 See Airy's Tract, Art. 163.



REFLEXION OF POLARIZED LIGHT. 255

have attempted in any published work to remove the
difficuUy, and it is not to be supposed that they had
despised it.

And by the analogy of certain geometrical cases where the multi-
plication by -v/ — 1 indicates a line differing in angular position by 90°,
he hazarded the inference that such an interpretation might hold
good here, and that this expression would be equivalent to one of the

form,

27r . 2/r

cos d sin -TT- (vl — x) + sm d sui — {vi—x + 90°;

A A

which is trigonometrically the same as

2n- , \



( 21" ,



1

This applying to the component in the plane of incidence, a similar

expression would apply to that perpendicular to it,

. / 27r \

or sm ( — - (vt — x) + dl )

The difference of these expressions, or the relative retardation of
the two sets of waves, will he d — 6' = 6-

In general, 6 having any value, and the plane of polarization being
inclined at an angle a to the plane of incidence on the rhomb, the
components are,

y = sin a sin — {vt — x + 6) (■*••)

A

z = cos a sin — (vt — x) (*•)

A ^ '

This then is precisely the same case as that considered in a former

note; and exactly in the same way we obtain,

ifl 22 itiz cos 6 . „ „

~~-\ 5 : — = sm 2(5.

sm 2« cos ^a cos a sm a

The general equation to an ellipse. If d = 90°, the semi-axes are
sin a and cos a, parallel and perpendicular to the plane of incidence.
If a =^ 45° and 6 variable, it is still an ellipse. If a = 45° and 6 = 90°,
it becomes a circle. Tims a ray polarized at an angle a, loith th^ plane
of incidence, after two internal reflexions in glass, emerges elliptically or
circularly polarized, according to the above condition.

From the empirical terms before mentioned, Fresnel derived ex-
pressions from which he calculated that for crown glass, where (i =
1'51, an internal incidence i = 54° 37' would give 6 = 45°. Thus
experimentally cutting a rhomb of such glass at that angle, so that
the ray polarized at 45° to the plane of incidence, entering one face



256 FRESNEL.

As to the system of waves, the interferences are so
natural a deduction from it, that we have some reason to
be astonished that experimenters should have discovered
them before theory had indicated them. To convince
ourselves of this, it sufi&ces to remark that a wave, in
propagating itself through an elastic medium, communi-
cates to the molecules of which it is composed an oscil-
latory motion, in virtue of which they displace themselves
successively in two opposite directions : this being under-
stood, it is evident that a series of waves will desti-oy
completely the effect of another series, if at every
point in the fluid the motion in one direction which the
first wave produces alone, shall coincide with the motion
in the opposite direction which would result from the
sole action of the other wave. The molecules solicited
at the same time by equal forces diametrically opposed,
will then remain at rest, for as long a period as they
would have freely oscillated if under the action of one
wave alone. Motion has desti-oyed motion ; now motion
is light.

I will not push further this enumeration, because we
can already judge on how many points the antagonists of
the emission theory have been successful in their attacks.
Experiments so numerous, so varied, so delicate, as those
I have referred to, do not alone testify all the importance
which the question seems to them to possess ; they must

perpendicularly, might be reflected internally at that angle, and,
passing to the opposite side, be reflected again internally at the same
angle ; after two reflexions it would emerge, consisting of two pencils
polarized at right angles to each other, and having a difi'erence of
phase 6 — 90°, and would thus possess a circular polarization ; or if
the inclination was any other than 45° and 6 diftering from 90, the
polarization would be elliptic of different degrees; all which conclu-
sions are fully verified by experiments as before noticed.



DIVERGENCE OF LIGHT. 257

be regarded further as a striking mark of respect towards
the great man whose name, so to speak, has been identi-
fied with the theory which they think ought to be re-
jected. As to the theory of waves, the Newtonians have
not done it the honour to discuss it with the same detail ;
it has seemed to them that a single objection was suffi-
cient to annihilate it ; and this objection they have drawn
from the manner in which sound is propagated in air. If
light, they say, is a vibration like the vibrations of sound,
it will be transmitted in all directions ; just as we hear
the sound of a distant bell when we are separated from it
by a screen which conceals it from our eyes, in the same
way we ought to perceive the light of the sun behind
every kind of opaque body. Such are the terms to
which we must reduce the difficulty, for analogy does not
permit us to say that light ought to extend itself behind
screens without losing some of its intensity ; since sound
itself, as every one knows, does not penetrate obstacles
without being enfeebled in a sensible degree. Thus, in
speaking of the extension of light into the geometrical
shadow of a body as an insurmountable difficulty, New-
ton and his adherents certainly did not suspect the answer
which it would bring with it ; yet this answer is direct
and simple. You maintain that the luminous vibrations
ought to extend into the shadow, — they do so. You say
that in the system of waves, the shadow of an opaque
body can never be comjiletely dark, — it nei^er is so. It
includes a number of rays which give rise to a multitude
of curious phenomena, of which you may have some
knowledge, since Grimaldi perceived them in part so
long ago as before 1633.* Fresnel, — and here is incon-

* Amoncf the earliest difficulties which seemed to attend the con-
ception of the wave theory, was the consideration, which appeared so



258 FRESNEL.

testably one of the most important of his discoveries, —
has shown how and under what circumstances this diver-

un answerable, that on this principle there ought to be no darkness ;
light ought to spread equally into the shadow, and we ought to see
round a corner.

It was the fertile principle of interference which was to supply the
answer, as indeed had been long before hinted generally by Huy-
ghens. The waves diverging from tlie different parts of a luminous
source of any sensible magnitude interfere with and neutralize each
other, except in the main direction, when alone they exactlj'- concur;
— a principle called "the mutual destruction of secondary waves."
Young dwelt much at first on this objection; and afterwards, in a
letter to Arago. he renews a similar expression of the difficulties he
felt in another point of view: " If light has so great a tendency to
diverge into the path of neighbouring rays, and to interfere with
them, as Huyghens supposed, I do not see how it escapes being to-
tally extinguished in a very short space, even in the most transparent
medium." — Peacock's Life, p. 140. But the principle just adverted
to shows that the middle portion of the light coming from a point of
any physical magnitude is not subject to those mutual interferences,
and does not diverge, but is perpetually reinforced by the supply of
fresh waves incessantly propagated from the original source. In
these explanations Young at length expressed his full concuiTence in
a letter to Fresnel. The actual divergence of light into a shadow is
demonstrated by the existence of the internal stripes. This, however,
is an effect only produced to a very limited extent; and the general
law of the "mutual destruction of secondary waves" in ordinary
cases applies to produce the effect of destroying all apparent lateral
divergence. There are, however, some cases where this cause operates
less extensively (such, at least, would seem to be the case, and is the
view upheld by some mathematicians); at all events, under certain
conditions, the divergence is rendered very much more conspicuous,
and reaches to a far greater distance from the edge. This appears to
have been the case in a remarkable experiment, mentioned both by
Newton and Hooke, and probably observed bj' each independently,
but described, especially by Newton, in somewhat obscure terms (see
Optics, book iii. part i. obs. 5, (Ed. 1721,) but more precisely by Hooke:
see Tosthumous Works, pp. 186 and 190, and plate 11, fig. 8, p. 155,
Ed. 1705). Hooke ascribes it to a "deflexion of light differing both
from reflexion and refraction, and seeming to depend on the unequal
density of the constituent parts of the ray," &c. Newton enters ou



DIVERGENCE OF LIGHT. 259

gence of light takes place : he has further shown that in
a complete wave which is freely propagated, the rays are

no theoretical considerations whatever, but mentions it only among
those unfinished inquiries which, as he says, he had left imperfect
and was unable to carry out.

Both the fact, and all questions relating to it, seem to have been
overlooked until, in reference to a somewhat similar case, M. Babinet
supposed that under particular conditions the mutual interference of
the secondary waves might be interrupted by stopping one of the in-
terfering portions of light, and thus the other portion be rendered
effective, and consequently diverging rays made visible. The author
of this note, in relation to what appears a closely allied, if not iden-
tical phenomenon, the formation of a corona or ring of light round the
dark disk of the moon in a total eclipse of the sun, tried some analo-
gous experiments, and rendered the same kind of efifect conspicuous
and easy to be studied by an arrangement of this kind: —

The rays of the sun Q are transmitted by reflexion from an inclined
mirror (in) through a small hole {h) in a shutter, and in the diverging
beam is placed an opaque circular disk (d) which intercepts the rays
at a point where they have an area considerably less than its own dia-
meter. From the edge of (d) i-ays are seen to diverge into its shadow
and cross at successive points along the axis; they are thus rendered

G




visible by means of a small eye lens at (e) which presents the appear-
ance of the shadow of the circular disk, having a multitude of rays
converging inwards from its edge to its centre, where they form a
point or small circle of great relative brightness. If, on the other
hand, the disk (rZ) under the same conditions be viewed directhj by
the eye, without the lens, its shadow is seen relatively and uniformly
dark, but sun-ounded by a bright luminous ring on its outside. The
same appearance of the ring is also presented if, instead of the solar
rays, we use the light of a flame placed at the principal focus of a lens
inserted in a screen so as to send oiit a beam of parallel pencils inter-
cepted in like manner by the disk. In this case, however, the con-



260 FRESNEL.

only sensible in the directions which, prolonged, ter-
minate in the luminous points, although in eacli of its
successive positions the different parts of the primitive
■wave are in fact themselves the centres of disturbance,
whence emanate new waves in all directions ; but these
oblique or secondary waves interfere with each other,
and destroy each other entirely. There remain then
only the normal waves ; and thus the rectilinear propa-
gation of light finds an explanation in the system of
vibrations.

"When the original wave is not entire, when it is broken
or intercepted by the presence of an opaque body, the
result of the interferences (which in this case play an
important part) is not so simple to explain : the rays
which go off obliquely from all parts of the wave not in-
tercepted, do not necessarily destroy each other. In one
part they conspire with the normal ray, and produce a
brilliant light ; in another these same rays destroy each
other, and all light disappears. From the point where a
ray is broken, its propagation is eftected thenceforward
according to special laws ; the light which falls upon a
screen is no longer uniform : it necessarily is composed
of alternate stripes of brightness and darkness regularly
placed. If the opaque intercepting body is not very
large, the oblique waves which cross each other within

verging rays cannot be seen. This apparently paradoxical effect has
been supposed by some not sufficiently explained on M. Babinet's
principle. The reader will find some observations on the subject, and
its applications in the author's two papers in the Memoirs of the Royal
Astronomical Society, vol. xvi., on Luminous Rings round Shadows,
and in vol. xviii., on Irradiation. Some further remarks also will be
found in his paper on Lord Brougham's Experiments, Phil. Mag. July,
1852. — Translator.



AVAVE THEORY. 2#1

its shadow produce, by their reciprocal action, stripes
analogous to the former, but differently distributed.

I perceive that, without intending it, in following the
theoretical speculations of Fresnel, I have mentioned the
principal features of those curious phenomena of diffrac-
tion, which I have before cited under another point of
view, to which Newton devoted one entire book of his
Optics. Newton believed that he could not give any ex-
planation of these phenomena (so difficult did they seem
to him), except by admitting that a ray of light cannot
pass close to a body without there undergoing a sinuous
movement like that of an eel. In the explanations of
Fresnel this strange supposition is superfluous.

The opaque body which seems to be the original cause
of the diffracted bands does not act at all on the rays,
either by attraction or by repulsion ; it simply intercepts
a part of the principal wave. It stops in the ratio of
their breadth a great number of oblique rays, which, but
for this interruption, would have gone into certain parts
of space to mix with other rays, and to interfere more or
less with them.

Thus it is no longer surprising that, as observation has
proved, the resulting effect is independent of the nature
and mass of the body. The periods of maximum and
minimum of the light, as well without as within the
shadow, are directly deducible from the theory of Fresnel
with a degree of precision of which hitherto, perhaps, no
branch of physical science had afforded so striking an
example. Thus, whatever reserve it may be prudent to
impose on ourselves when we run the risk of speaking
of the labours of our successors, I would almost venture
to affirm that, with regard to diffraction, they will add
nothing essential to the discoveries with which Fresnel



262 FRESNEL.

has enriched the science. Theories lire, in general, only
metliods, more or less happy, of linking together a certain
number of facts already known. But when all the new
consequences which we can deduce fi'ora them are found
to agree with experience, they claim a higher importance.
This kind of success has not been wanting to Fresnel.
His formulas of diffraction include, by implication, a very
sti-ange result, which he had not perceived. One of our
colleagues* — I shall have no need to mention his name,
if 1 say that he has been placed long since among the
greatest geometers of this age, as well by a multitude of
important labours in pure analysis, as by the most happy
applications to the system of the world, and to physics, —
perceived at a glance the consequence of which I have
spoken ; he showed that, in admitting the formulas of
Fresnel, the centre of the shadow of an opaque and cir-
cular screen ought to be as bright as if the screen did not
exist. This consequence, apparently so paradoxical, was
subjected to trial by direct experiment, and observation
has perfectly confirmed the result of calculation.

In the long and difficult discussion to which the nature
of light has given birth, and of which I have just traced
the history, the task of the physicists has been nearly ful-
filled ; as to that of the mathematicians, it unhappily still
offers some deficiencies to be filled up. I would venture
then, if I had the right, to adjure that great geometer
to whom optical science owes the important result just
mentioned, to try whether the half empirical formulas by
which Fresnel has attempted to express the intensities
of light reflected under all angles and for all kinds of
surfaces, may not be found deducible also from the gen-
eral equations of motion of elastic fluids. It remains,
* Poisson,



LIGHT-HOUSES. 263

above all, to explain how the difFererit undulations can
undergo unequal deviations at the bounding t^urfaces of
transparent bodies.

LIGHT-HOUSES.

In an academy of sciences, if it properly appreciate its
functions, the author of a discovery is never exposed to
the discouraging question so often addressed to him in
the world, of cui bono ? Here every one comprehends
that the animal life ought not to be the sole occupation of
man ; that the cultivation of his intellect, — that an atten-
tive study of this infinite variety of animated beings, and
inert matter, with which he is surrounded, forms the most
beautiful portion of his destined pursuits.

But besides, even if we were desirous to find nothing
in the sciences but the means of facilitating the reproduc-
tion of substances for food, — of weaving with more or less
economy and perfection the different fabrics which serve
for clothing, — of constructing with elegance and solidity
the convenient habitations in which we escape the vicis-
situdes of the seasons, — of extracting from the bowels of
the earth so many metals and combustible matter, which
are necessary for the arts of life, — of annihilating a hun-
dred material obstacles which oppose themselves to the
intercommunication of inhabitants of the same continent,
of the same kingdom, even of the same city, — of extract-
ing and preparing the medicaments proper for combating
the numerous disorders with which our organs are inces-
santly threatened, — the question of cui bono 2 will be
found completely announced. Natural phenomena have
innumerable points of connection with each other, often
hidden, the discovery of which one age bequeathes to
another. At the moment when these relations ai'e dis-



264 FEESNEL.



covered, important applications rise up, as if by enchant-
ment, out of experiments wliich, until then, would seem
likely to remain for ever among the number of abstract
speculations. A fact which no direct utility had as yet
recommended to the attention of the public becomes, per-
haps, the step on which a man of genius supports himself
to climb up to those primary truths which change the
■whole face of science, whether for creation of some eco-
nomical moving power, which all manufacturing arts will
henceforth adopt, and of which not the least merit is that
of delivering thousands of operatives from overwhelming
toils which assimilated them with the brutes, ruined their
health, and brought them to a premature death. If to
fortify these reflections examples may be thought neces-
sary, I should feel no other embarrassment than that of
too wide a choice. But here there is no necessity to
enter on such details ; for to all the theoretical researches
already mentioned, Fresnel has added an important labour,
having an immediate practical application, which will cer-
tainly place his name among those of the benefactors of
the human race. This work, every one knows, had for
its object the improvement of light- houses. I will pro-
ceed to trace the outline of its progress, and shall thus
have finished the sketch which I proposed to offer you of
the brilliant scientific career of our late colleague.

Persons unacquainted with nautical matters are usually
seized with a sort of fear when the vessel which carries
them, at a distance from continents or islands, has no
other witness of its progress than the stars and the waves.
A view of any coast the most barren, the most rocky, the
most inhospitable, dissipates, as if by enchantment, those
undefined fears which their absolutely isolated position
had inspired, while, to the experienced navigator, it is



Online LibraryF. (François) AragoBiographies of distinguished scientific men (Volume 2) → online text (page 20 of 38)