have been subjected to the closest examination. They do not behave
alike. One of them obeys the ordinary law of refraction discovered by
Snell, and is, therefore, called the _ordinary ray_: its index of
refraction is 1.654. The other does not obey this law. Its index of
refraction, for example, is not constant, but varies from a maximum of
1.654 to a minimum of 1.483; nor in this case do the incident and
refracted rays always lie in the same plane. It is, therefore, called
the _extraordinary ray_. In calc-spar, as just stated, the ordinary
ray is the most refracted. One consequence of this merits a passing
notice. Pour water and bisulphide of carbon into two cups of the same
depth; the cup that contains the more strongly refracting liquid will
appear shallower than the other. Place a piece of Iceland spar over a
dot of ink; two dots are seen, the one appearing nearer than the other
to the eye. The nearest dot belongs to the most strongly refracted
ray, exactly as the nearest cup-bottom belongs to the most highly
refracting liquid. When you turn the spar round, the extraordinary
image of the dot rotates round the ordinary one, which remains fixed.
This is also the deportment of our two disks upon the screen.
§ 5. _Polarization of Light explained by the Wave Theory_.
The double refraction of Iceland spar was first treated in a work
published by Erasmus Bartholinus, in 1669. Huyghens sought to account
for this phenomenon on the principles of the wave theory, and he
succeeded in doing so. He, moreover, made highly important
observations on the distinctive character of the two beams transmitted
by the spar, admitting, with resigned candour, that he had not solved
the difficulty, and leaving the solution to future times. Newton,
reflecting on the observations of Huyghens, came to the conclusion
that each of the beams transmitted by Iceland spar had two sides; and
from the analogy of this _two-sidedness_ with the _two-endedness_ of a
magnet, wherein consists its polarity, the two beams came subsequently
to be described as _polarized_.
We may begin the study of the polarization of light, with ease and
profit, by means of a crystal of tourmaline. But we must start with a
clear conception of an ordinary beam of light. It has been already
explained that the vibrations of the individual ether-particles are
executed _across_ the line of propagation. In the case of ordinary
light we are to figure the ether-particles as vibrating in all
directions, or azimuths, as it is sometimes expressed, across this
line.
Now, in the case of a plate of tourmaline cut parallel to the axis of
the crystal, a beam of light incident upon the plate is divided into
two, the one vibrating parallel to the axis of the crystal, the other
at right angles to the axis. The grouping of the molecules, and of
the ether associated with the molecules, reduces all the vibrations
incident upon the crystal to these two directions. One of these beams,
namely, that whose vibrations are perpendicular to the axis, is
quenched with exceeding rapidity by the tourmaline. To such vibrations
many specimens of the crystal are highly opaque; so that, after having
passed through a very small thickness of the tourmaline, the light
emerges with all its vibrations reduced to a single plane. In this
condition it is what we call _plane polarized light_.
[Illustration: Fig. 27.]
[Illustration: Fig. 28.]
A moment's reflection will show that, if what is here stated be
correct, on placing a second plate of tourmaline with its axis
parallel to the first, the light will pass through both; but that, if
the axes be crossed, the light that passes through the one plate will
be quenched by the other, a total interception of the light being the
consequence. Let us test this conclusion by experiment. The image of a
plate of tourmaline (_t_ _t_, fig. 27) is now before you. I place
parallel to it another plate (_t'_ _t'_): the green of the crystal is
a little deepened, nothing more; this agrees with our conclusion. By
means of an endless screw, I now turn one of the crystals gradually
round, and you observe that as long as the two plates are oblique to
each other, a certain portion of light gets through; but that when
they are at right angles to each other, the space common to both is a
space of darkness (fig. 28). Our conclusion, arrived at prior to
experiment, is thus verified.
Let us now return to a single plate; and here let me say that it is on
the green light transmitted by the tourmaline that you are to fix your
attention. We have to illustrate the two-sidedness of that green
light, in contrast to the all-sidedness of ordinary light. The white
light surrounding the green image, being ordinary light, is reflected
by a plane glass mirror in all directions; the green light, on the
contrary, is not so reflected. The image of the tourmaline is now
horizontal; reflected upwards, it is still green; reflected sideways,
the image is reduced to blackness, because of the incompetency of the
green light to be reflected in this direction. Making the plate of
tourmaline vertical, and reflecting it as before, it is the light of
the upper image that is quenched; the side image now shows the green.
This is a result of the greatest significance. If the vibrations of
light were longitudinal, like those of sound, you could have no action
of this kind; and this very action compels us to assume that the
vibrations are transversal. Picture the thing clearly. In the one case
the mirror receives, as it were, the impact of the _edges_ of the
waves, the green light being then quenched. In the other case the
_sides_ of the waves strike the mirror, and the green light is
reflected. To render the extinction complete, the light must be
received upon the mirror at a special angle. What this angle is we
shall learn presently.
The quality of two-sidedness conferred upon light by bi-refracting
crystals may also be conferred upon it by ordinary reflection. Malus
made this discovery in 1808, while looking through Iceland spar at the
light of the sun reflected from the windows of the Luxembourg palace
in Paris. I receive upon a plate of window-glass the beam from our
lamp; a great portion of the light reflected from the glass is
polarized. The vibrations of this reflected beam are executed, for the
most part, parallel to the surface of the glass, and when the glass is
held so that the beam shall make an angle of 58° with the
perpendicular to the glass, the _whole_ of the reflected beam is
polarized. It was at this angle that the image of the tourmaline was
completely quenched in our former experiment. It is called _the
polarizing angle_.
Sir David Brewster proved the angle of polarization of a medium to be
that particular angle at which the refracted and reflected rays
inclose a right angle.[17] The polarizing angle augments with the
index of refraction. For water it is 52½°; for glass, as already
stated, 58°; while for diamond it is 68°.
And now let us try to make substantially the experiment of Malus. The
beam from the lamp is received at the proper angle upon a plate of
glass and reflected through the spar. Instead of two images, you see
but one. So that the light, when polarized, as it now is by
reflection, can only get through the spar in one direction, and
consequently can produce but one image. Why is this? In the Iceland
spar as in the tourmaline, all the vibrations of the ordinary light
are reduced to two planes at right angles to each other; but, unlike
the tourmaline, both beams are transmitted with equal facility by the
spar. The two beams, in short, emergent from the spar, are polarized,
their directions of vibration being at right angles to each other.
When, therefore, the light is first polarized by reflection, the
direction of vibration in the spar which coincides with the direction
of vibration of the polarized beam, transmits the beam, and that
direction only. Only one image, therefore, is possible under the
conditions.
You will now observe that such logic as connects our experiments is
simply a transcript of the logic of Nature. On the screen before you
are two disks of light produced by the double refraction of Iceland
spar. They are, as you know, two images of the aperture through which
the light issues from the camera. Placing the tourmaline in front of
the aperture, two images of the crystal will also be obtained; but now
let us reason out beforehand what is to be expected from this
experiment. The light emergent from the tourmaline is polarized.
Placing the crystal with its axis horizontal, the vibrations of its
transmitted light will be horizontal. Now the spar, as already stated,
has two directions of vibration, one of which at the present moment
is vertical, the other horizontal. What are we to conclude? That the
green light will be transmitted along the latter, which is parallel to
the axis of the tourmaline, and not along the former, which is
perpendicular to that axis. Hence we may infer that one image of the
tourmaline will show the ordinary green light of the crystal, while
the other image will be black. Tested by experiment, our reasoning is
verified to the letter (fig. 29).
[Illustration: Fig. 29.]
[Illustration; Fig. 30.]
Let us push our test still further. By means of an endless screw, the
crystal can be turned ninety degrees round. The black image, as I
turn, becomes gradually brighter, and the bright one gradually darker;
at an angle of forty-five degrees both images are equally bright (fig.
30); while, when ninety degrees have been obtained, the axis of the
crystal being then vertical, the bright and black images have changed
places, exactly as reasoning would have led us to suppose (fig. 31).
[Illustration: Fig. 31.]
[Illustration: Fig. 32.]
Considering what has been already said (p. 114) regarding the
reflection of light polarized by transmission through tourmaline, you
will readily foresee what must occur when we receive upon a plate of
glass, held at the polarizing angle, the two beams emergent from our
prism of Iceland spar. I cause both beams to pass side by side through
the air, catch them on a glass plate, and seek to reflect them
upwards. At the polarizing angle one beam only is capable of being
thus reflected. Which? Your prompt answer will be, The beam whose
vibrations are horizontal (fig. 32). I now turn the glass plate and
try to reflect both beams laterally. One of them only is reflected;
that, namely, the vibrations of which are vertical (fig. 33). It is
plain that, by means either of the tourmaline or the reflecting glass,
we can determine in a moment the direction of vibration in any
polarized beam.
[Illustration: Fig. 33.]
As already stated, the whole of a beam of ordinary light reflected
from glass at the polarizing angle is polarized; a word must now be
added regarding the far larger portion of the light which is
_transmitted_ by the glass. The transmitted beam contains a quantity
of polarized light equal to the reflected beam; but this is only a
fraction of the whole transmitted light. By taking two plates of glass
instead of one, we augment the quantity of the transmitted polarized
light; and by taking _a bundle_ of plates, we so increase the quantity
as to render the transmitted beam, for all practical purposes,
_perfectly_ polarized. Indeed, bundles of glass plates are often
employed as a means of furnishing polarized light. It is important to
note that the plane of vibration of this transmitted light is at right
angles to that of the reflected light.
One word more. When the tourmalines are crossed, the space where they
cross each other is black. But we have seen that the least obliquity
on the part of the crystals permits light to get through both. Now
suppose, when the two plates are crossed, that we interpose a third
plate of tourmaline between them, with its axis oblique to both. A
portion of the light transmitted by the first plate will get through
this intermediate one. But, after it has got through, _its plane of
vibration is changed_: it is no longer perpendicular to the axis of
the crystal in front. Hence it will, in part, get through that
crystal. Thus, by pure reasoning, we infer that the interposition of a
third plate of tourmaline will in part abolish the darkness produced
by the perpendicular crossing of the other two plates. I have not a
third plate of tourmaline; but the talc or mica which you employ in
your stoves is a more convenient substance, which acts in the same
way. Between the crossed tourmalines, I introduce a film of this
crystal with its axis oblique to theirs. You see the edge of the film
slowly descending, and, as it descends, light takes the place of
darkness. The darkness, in fact, seems scraped away, as if it were
something material. This effect has been called, naturally but
improperly, _depolarization_. Its proper meaning will be disclosed in
our next lecture.
These experiments and reasonings, if only thoroughly studied and
understood, will form a solid groundwork for the analysis of the
splendid optical phenomena next to be considered.
LECTURE IV.
CHROMATIC PHENOMENA PRODUCED BY CRYSTALS IN POLARIZED LIGHT
THE NICOL PRISM
POLARIZER AND ANALYZER
ACTION OF THICK AND THIN PLATES OF SELENITE
COLOURS DEPENDENT ON THICKNESS
RESOLUTION OF POLARIZED BEAM INTO TWO OTHERS BY THE SELENITE
ONE OF THEM MORE RETARDED THAN THE OTHER
RECOMPOUNDING OF THE TWO SYSTEMS OF WAVES BY THE ANALYZER
INTERFERENCE THUS RENDERED POSSIBLE
CONSEQUENT PRODUCTION OF COLOURS
ACTION OF BODIES MECHANICALLY STRAINED OR PRESSED
ACTION OF SONOROUS VIBRATIONS
ACTION OF GLASS STRAINED OR PRESSED BY HEAT
CIRCULAR POLARIZATION
CHROMATIC PHENOMENA PRODUCED BY QUARTZ
THE MAGNETIZATION OF LIGHT
RINGS SURROUNDING THE AXES OF CRYSTALS
BIAXAL AND UNIAXAL CRYSTALS
GRASP OF THE UNDULATORY THEORY
THE COLOUR AND POLARIZATION OF SKY-LIGHT
GENERATION OF ARTIFICIAL SKIES.
§ 1. _Action of Crystals on Polarized Light: the Nicol Prism._
We have this evening to examine and illustrate the chromatic phenomena
produced by the action of crystals, and double-refracting bodies
generally, upon polarized light, and to apply the Undulatory Theory to
their elucidation. For a long time investigators were compelled to
employ plates of tourmaline for this purpose, and the progress they
made with so defective a means of inquiry is astonishing. But these
men had their hearts in their work, and were on this account enabled
to extract great results from small instrumental appliances. For our
present purpose we need far larger apparatus; and, happily, in these
later times this need has been to a great extent satisfied. We have
seen and examined the two beams emergent from Iceland spar, and have
proved them to be polarized. If, at the sacrifice of half the light,
we could abolish one of these, the other would place at our disposal a
beam of polarized light, incomparably stronger than any attainable
from tourmaline.
The beams, as you know, are refracted differently, and from this, as
made plain in §4, Lecture I., we are able to infer that the one may be
totally reflected, when the other is not. An able optician, named
Nicol, cut a crystal of Iceland spar in two halves in a certain
direction. He polished the severed surfaces, and reunited them by
Canada balsam, the surface of union being so inclined to the beam
traversing the spar that the ordinary ray, which is the most highly
refracted, was totally reflected by the balsam, while the
extraordinary ray was permitted to pass on.
Let _b x, c y_ (fig. 34) represent the section of an elongated rhomb
of Iceland spar cloven from the crystal. Let this rhomb be cut along
the plane _b c_; and the two severed surfaces, after having been
polished, reunited by Canada balsam. We learned, in our first lecture,
that total reflection only takes place when a ray seeks to escape from
a more refracting to a less refracting medium, and that it always,
under these circumstances, takes place when the obliquity is
sufficient. Now the refractive index of Iceland spar is, for the
extraordinary ray less, and for the ordinary greater, than for Canada
balsam. Hence, in passing from the spar to the balsam, the
extraordinary ray passes from a less refracting to a more refracting
medium, where total reflection cannot occur; while the ordinary ray
passes from a more refracting to a less refracting medium, where
total reflection can occur. The requisite obliquity is secured by
making the rhomb of such a length that the plane of which _b c_ is the
section shall be perpendicular, or nearly so, to the two end surfaces
of the rhomb _b x, c y_.
[Illustration: Fig. 34.]
The invention of the Nicol prism was a great step in practical optics,
and quite recently such prisms have been constructed of a size and
purity which enable audiences like the present to witness the
chromatic phenomena of polarized light to a degree altogether
unattainable a short time ago.
(The two prisms employed in these experiments were lent to me by my
lamented friend Mr. William Spottiswoode, and they were manufactured
by Mr. Ahrens, an optician of consummate skill.)
§ 2. _Colours of Films of Selenite in Polarized Light_.
Two Nicol prisms play the same part as the two plates of tourmaline.
Placed with their directions of vibration parallel, the light passes
through both; while when these directions are crossed the light is
quenched. Introducing a film of mica between the prisms, the light, as
in the case of the tourmaline, is restored. But notice, when the film
of mica is _thin_ you have sometimes not only light, but _coloured_
light. Our work for some time to come will consist of the examination
of such colours. With this view, I will take a representative crystal,
one easily dealt with, because it cleaves with great facility--the
crystal gypsum, or selenite, which is crystallized sulphate of lime.
Between the crossed Nicols I place a thick plate of this crystal; like
the mica, it restores the light, but it produces no colour. With my
penknife I take a thin splinter from the crystal and place it between
the prisms; the image of the splinter glows with the richest colours.
Turning the prism in front, these colours gradually fade and
disappear, but, by continuing the rotation until the vibrating
sections of the prisms are parallel to each other, vivid colours again
arise, but these colours are complementary to the former ones.
Some patches of the splinter appear of one colour, some of another.
These differences are due to the different thicknesses of the film. As
in the case of Hooke's thin plates, if the thickness be uniform the
colour is uniform. Here, for instance, is a stellar shape, every
lozenge of the star being a film of gypsum of uniform thickness: each
lozenge, you observe, shows a brilliant and uniform colour. It is
easy, by shaping our films so as to represent flowers or other
objects, to exhibit such objects in hues unattainable by art. Here,
for example, is a specimen of heart's-ease, the colours of which you
might safely defy the artist to reproduce. By turning the front Nicol
90 degrees round, we pass through a colourless phase to a series of
colours complementary to the former ones. This change is still more
strikingly represented by a rose-tree, which is now presented in its
natural hues--a red flower and green leaves; turning the prism 90
degrees round, we obtain a green flower and red leaves. All these
wonderful chromatic effects have definite mechanical causes in the
motions of the ether. The principle of interference duly applied and
interpreted explains them all.
§ 3. _Colours of Crystals in Polarized Light explained by the
Undulatory Theory_.
By this time you have learned that the word 'light' may be used in two
different senses: it may mean the impression made upon consciousness,
or it may mean the physical cause of the impression. It is with this
cause that we have to occupy ourselves at present. The luminiferous
ether is a substance which fills all space, and surrounds the atoms
and molecules of bodies. To this inter-stellar and inter-atomic medium
definite mechanical properties are ascribed, and we deal with it in
our reasonings and calculations as a body possessed of these
properties. In mechanics we have the composition and resolution of
forces and of motions, extending to the composition and resolution of
_vibrations_. We treat the luminiferous ether on mechanical
principles, and, from the composition and resolution of its
vibrations we deduce all the phenomena displayed by crystals in
polarized light.
[Illustration: Fig. 35.]
Let us take, as an example, the crystal of tourmaline, with which we
are now so familiar. Let a vibration cross this crystal oblique to its
axis. Experiment has assured us that a portion of the light will pass
through. The quantity which passes we determine in this way. Let A B
(fig. 35) be the axis of the tourmaline, and let _a_ _b_ represent the
amplitude of an oblique ethereal vibration before it reaches A B. From
_a_ and _b_ let the two perpendiculars _a_ _c_ and _b_ _d_ be drawn
upon the axis: then _c_ _d_ will be the amplitude of the transmitted
vibration.
I shall immediately ask you to follow me while I endeavour to explain
the effects observed when a film of gypsum is placed between the two
Nicol prisms. But, prior to this, it will be desirable to establish
still further the analogy between the action of the prisms and that of
the two plates of tourmaline. The magnified images of these plates,
with their axes at right-angles to each other, are now before you.
Introducing between them a film of selenite, you observe that by
turning the film round it may be placed in a position where it has no
power to abolish the darkness of the superposed portions of the
tourmalines. Why is this? The answer is, that in the gypsum there are
two directions, at right angles to each other, in which alone
vibrations can take place, and that in our present experiment one of
these directions is parallel to one of the axes of the tourmaline, and
the other parallel to the other axis. When this is the case, the film
exercises no sensible action upon the light. But now I turn the film
so as to render its directions of vibration _oblique_ to the two
tourmaline axes; then, you see it exercises the power, demonstrated in
the last lecture, of partially restoring the light.
[Illustration: Fig. 36.]
Let us now mount our Nicol prisms, and cross them as we crossed the
tourmaline. Introducing our film of gypsum between them, you notice
that in one particular position the film has no power whatever over
the field of view. But, when the film is turned a little way round,
the light passes. We have now to understand the mechanism by which
this is effected.
First, then, we have a prism which receives the light from the
electric lamp, and which is called the _polarizer_. Then we have the
plate of gypsum (supposed to be placed at S, fig. 36), and then the
prism in front, which is called the _analyzer_. On its emergence from
the first prism, the light is polarized; and, in the particular case
now before us, its vibrations are executed in a horizontal plane. We
have to examine what occurs when the two directions of vibration in
the interposed gypsum are oblique to the horizon. Draw a rectangular
cross (A B, C D, fig. 37) to represent these two directions. Draw a
line (_a_ _b_) to represent the amplitude of the horizontal vibration
on the emergence of the light from the first Nicol. Let fall from each
end of this line two perpendiculars (_a_ _c_, _a_ _f_, _b_ _d_, _b_
_e_) on the two arms of the cross; then the distances (_c_ _d_, _e_
_f_) between the feet of these perpendiculars represent the amplitudes
of two rectangular vibrations, which are the _components_ of the first
single vibration. Thus the polarized ray, when it enters the gypsum,
is resolved into its two equivalents, which vibrate at right angles to
each other.
[Illustration; Fig. 37.]