when the media through which it passes are uniform; for it is easily
proved that when light passes from one medium to another, a change of
direction takes place at the common surface of the media in all rays
that meet this surface otherwise than perpendicularly. As a consequence
of this, it really is possible to see round a corner, as the reader may
convince himself by performing the following easy experiment. Having
procured a cup or basin, Fig. 202, let him, by means of a little
bees’-wax or tallow, attach to the bottom of the vessel, at R, a small
coin. If he now places the cup so that its edge just conceals the coin
from view, and maintains his eye steadily in the same position as at I,
he will, when water is poured into the cup, perceive the coin apparently
above the edge of the vessel in the direction I R´, that is, the bottom
of the cup will appear to have risen higher. Since it is known that in
each medium the rays pass in straight lines, the bending which renders
the coin visible can therefore only take place at the common junction of
the media, or, in other words, the ray, R O, passing from the object in
a straight line through the water, is bent abruptly aside as it passes
out at the surface of the water, A B, and enters the air, in which it
again pursues a straight course, reaching the eye at I, where it gives
the spectator an impression of an object at R´. This experiment is also
an illustration of the cause of the well-known tendency we have to
under-estimate the depth of water when we can see the bottom. The broken
appearance presented by an oar plunged into clear water is due to
precisely the same cause. The curious exaggerated sizes and distorted
shapes of the gold-fish seen in a transparent globe have their origin in
the same bending aside of the rays. This deviation which light undergoes
in passing obliquely from one medium into another is known by the name
of _refraction_, and it is essential for the understanding of the sequel
that the reader should be acquainted with some of the laws of this
phenomenon, although their discovery by Snell dates two centuries and a
half anterior to the present time. Let T O, Fig. 203, be a ray of light
which falls obliquely upon a plane surface, A B, common to two different
media, one of which is represented by the shaded portion of the figure,
A B C D, of which C D represents another plane surface, parallel to the
former. If the ray, T O, suffered no refraction, it would pursue its
course in a straight line to _R´_; but as a matter of fact it is found
that such a ray is always bent aside at O, if the medium A B C D is more
or less dense than the other. If, for example, A B C D is water, and the
medium above it glass, then the ray entering at O will take the course O
_R_; but if A B C D is a plate of glass with water above and below it,
the ray will take the course T O, O R, R B, suffering refraction on
entering the glass, and again on leaving it, so that R B will emerge
from the glass parallel to its original direction at T O. If through the
point of incidence, O, we suppose a line, O P, to be drawn perpendicular
to the surface, A B, then we may say that the ray in passing from the
rarer medium (water, air, &c.) into the denser medium (glass, &c.) is
bent towards the perpendicular, or normal, as at O; but that on leaving
the denser to enter the rarer medium, as at R, it is bent away from the
perpendicular. In other words, the angle _b_ O _a_ is less than the
angle _m_ O T, and O R forms a less angle with R P´ than R B´ does. It
is also a law of _ordinary_ refraction that the normal, O P, at the
point of incidence, the incident ray, T O, and the refracted ray, O R,
are all in the same plane. Besides, there is the important and
interesting law discovered by Snell and by Descartes, which may thus be
explained with reference to Fig. 203. On the incident and refracted
rays, T O and O R, let us suppose that any equal distances, O _d_ and O
_b_, are measured off from O, and that from each of the points _a_ and
_b_, perpendiculars, _a m_ and _b n_, are drawn to the normal, P P,
which passes through O; then it is found that, whatever may be the angle
of incidence, T O P, or however it is made to vary, the length of the
line _a m_ bears always the same proportion to the line _b n_ for the
same two media. Thus, if A B C D be water, and T O enters it out of the
air, the length of the line _a m_ divided by the length of the line _a
b_ will always (whatever slope T O may have) give the quotient 1·33.
This number is, therefore, a constant quantity for air and water, and is
called the index of refraction for air into water. The law just
explained is expressed by the language of mathematics thus: For two
given media the ratio of the sines of the angles of incidence and of
refraction is constant.
[Illustration:
FIG. 202.
]
[Illustration:
FIG. 203.
]
It is an axiom in optical science that a ray of light when sent in the
opposite direction will pursue the same path. Thus in Fig. 203 the
direction of the light is represented as from T towards B´; but if we
suppose B´ R to be an incident ray, it would pursue the path B´ R, R O,
O T, and in passing out of the denser medium, A B C D at O, its
direction is farther from the normal, P P, or O T, as the law of sines,
_a m_ will be always longer than _n b_, and will bear a constant ratio
to it. Suppose the angle R O P to increase, then P O B will become a
right angle; that is, the emergent ray, O T, will just graze the
surface, A B, when the angle R O P has some definite value. If this last
angle be further increased, _no light at all will pass out of the
medium_ A B C D, but the ray R O will be totally reflected at O back
into the medium, A B C D, according to the laws of reflection. The angle
which R O forms with O P when O T just skims the surface, A B, is termed
the _limiting angle_, or the _critical angle_, and its value varies with
the media. The reader may easily see the total reflection in an
aquarium, or even in a tumbler of water, when he looks up through the
glass at the surface of the water, which has then all the properties of
a perfect mirror.
The power of lenses to form images of objects is entirely due to these
laws of refraction. The ordinary double-convex lens, for example, having
its surfaces formed of portions of spheres, refracts the rays so that
_all_ the rays which from _one luminous point_ fall upon the lens, meet
together again at a point on the other side, the said point being termed
their _focus_. It is thus that _images_ of luminous bodies are formed by
lenses. An explanation of the construction and theory of lenses cannot,
however, be entered into in this place.
One important remark remains to be made—namely, that in the above
statement of the laws of reflection and refraction, certain limitations
and conditions under which they are true and perfectly general have not
been expressed; for the mention of a number of particulars, which the
reader would probably not be in a condition to understand, would only
tend to confuse, and the explanation of them would lead us beyond our
limits. Some of these conditions belong to the phenomena we have to
describe, and are named in connection with them, and others, which are
not in immediate relation to our subject, we leave the reader to find
for himself in any good treatise on optics.
_DOUBLE REFRACTION AND POLARIZATION._
About two hundred years ago, a traveller, returning from Iceland,
brought to Copenhagen some crystals, which he had obtained from the Bay
of Roërford, in that island. These crystals, which are remarkable for
their size and transparency, were sent by the traveller to his friend,
Erasmus Bartholinus, a medical man of great learning, who examined them
with great interest, and was much surprised by finding that all objects
viewed through them appeared double. He published an account of this
singular circumstance in 1669, and by the discovery of this property of
Iceland spar, it became evident that the theory of refraction, the laws
of which had been studied by Snell and by Huyghens a few years before,
required some modification, for these laws required only one refracted
ray, and Iceland spar gave two. Huyghens studied the subject afresh, and
was able, by a geometrical conception, to bring the new phenomena within
the general theory of light. Iceland spar is chemically carbonate of
lime (calcium carbonate), and hence is also called calc spar, and, from
the shape of the crystals, it has also been termed rhombohedral spar.
The form in which the crystals actually present themselves is seen in
Fig. 204, which also represents the phenomenon of double refraction.
Iceland spar splits up very readily, but only along certain definite
directions, and from such a piece as that represented in Fig. 204 a
perfect rhombohedron, such as that shown in Fig. 206, is readily
obtained by cleavage; and then we have a solid having six lozenge-shaped
sides, each lozenge or side having two obtuse angles of 101° 55´, and
two acute angles of 78° 5´. Of the eight solid corners, such as A B C,
&c., six are produced by the meeting of one obtuse and two acute angles,
and _the remaining two solid corners are formed by the meeting of three
obtuse angles_. Let us imagine that a line is drawn from one of these
angles to the other: the diagonal so drawn forms the _optic axis_ of the
crystal, and a plane passing through the optic axis, A B, Fig. 205, and
through the bisectors of the angles, E A D and F B G, marks a certain
definite direction in the crystal, to which also belong all planes
parallel to that just indicated. Any one of such planes forms what is
termed a “principal section,” to which we shall presently refer.
[Illustration:
FIG. 204.
]
[Illustration:
FIG. 205.
]
[Illustration:
FIG. 206.
]
It will be observed that in Fig. 204 the white circle on a black ground
seen through the crystal is doubled; but that, instead of being white as
the circle really is, the images appear grey, except where they overlap,
and there the full whiteness is seen. If we place the crystal upon a dot
made on a sheet of paper, or having made a small hole with a pin in a
piece of cardboard, hold this up to the light, and place the crystal
against it, we see apparently two dots or two holes. The two images
will, if the dot or hole be sufficiently small, appear entirely detached
from each other. Now, if, keeping the face of the crystal against the
cardboard or paper, the observer turn the crystal round, he will see one
of the images revolve in a circle round the other, which remains
stationary. The latter is called the _ordinary_ image, and the former
the _extraordinary_ image. Let us place the crystal upon a straight
black line ruled on a horizontal sheet of paper, Fig. 205, and let us
suppose, in order to better define the appearance, that we place it so
that the _optic axis_, A B, is in a plane perpendicular to the paper, A
being one of the two corners where the three obtuse angles meet, and B
the other, and the face, A B C D, parallel to E G H B, which touches the
paper. Then, according to the laws of ordinary refraction, if we look
_straight_ down upon the crystal, we should see through it the line I K,
unchanged in position—that is, the ray would pass perpendicularly
through the crystal as shown by L M—and, in fact, a part of the ray does
this, and gives us the _ordinary_ image, O O´; but another part of the
ray departs from the laws of Snell and Descartes, and, following the
course L N Y´, enters the eye in the direction N Y´, producing the
impression of another line at L´, which is the _extraordinary_ ray, E
E´. If the crystal be turned round on the paper, E E´ will gradually
approach O O´, and the two images will coincide when the _principal
section_ is parallel to the line I K; but the coincidence is only
apparent, and results from the superposition of the two images—for a
mark placed on the line drawn on the paper will show two images, one of
which will follow the rotation of the crystal, and show itself to the
right or left of the _ordinary_ image, according as C is to the right or
left of A. So that there are really in every portion of the crystal two
images on the line, one of which turns round the other, and the
coalescence of the two images twice in each revolution is only apparent,
for the different parts of the lengths of the images do not coincide. On
continuing the revolution of the crystal after they apparently coincide,
the images are again seen to separate, the _extraordinary_ one being now
displaced on the other side, or always towards the point, C. Thus, then,
the ray, on entering the crystal, bifurcates, one branch passing through
the crystal and out of it in the same straight line, just as it would in
passing through a piece of glass, while the other is refracted at its
entrance into the crystal, although falling perpendicularly upon its
face, and again at its exit. And again, when a beam of light, R _r_,
Fig. 206, falls obliquely on a crystal of Iceland spar, it divides at
the face of the crystal into two rays, _R_ O, and _r_ E; the former,
which is the ordinary ray, follows the laws of ordinary refraction—it
lies in the plane of incidence, and obeys the law of sines, just as if
it passed through a piece of plate-glass. The _extraordinary_ ray, on
the other hand, departs from the plane of incidence, except when the
latter is parallel to the _principal section_, and the ratio of the
sines of the angles of incidence and refraction varies with the
incidence. The reader who is desirous of studying these curious
phenomena of _double_ refraction, and those of polarization, is strongly
recommended to procure some fragments of Iceland spar, which he can very
easily cleave into rhombohedra, and with these, which need not exceed
half an inch square, or cost more than a few pence, he can demonstrate
for himself the phenomena, and become familiar with their laws. He will
find very convenient the simple plan recommended by the Rev. Baden
Powell, of fixing one of the crystals to the inside of the lid of a
pill-box, through which a small hole has been made, and through the hole
and the crystal view a pin-hole in the bottom of the box, turning the
lid, and the crystal with it, to observe the rotation of the image. The
same arrangement will serve, by merely attaching another rhomb of spar
within the box, to study the very interesting facts of the polarization
to which we are about to claim the reader’s attention.
The curious phenomena which have just been described, although in
themselves by no means recent discoveries, have led to some of the most
interesting and beautiful results in the whole range of physical
science. The examination and discussion of them by such able
investigators as Huyghens, Descartes, Newton, Fresnel, Malus, and
Hamilton, have largely conduced to the establishment of the undulatory
hypothesis—that comprehensive theory of light, which brings the whole
subject within the reach of a few simple mechanical conceptions.
It was at first supposed that it was only one of the rays which are
produced in double refraction that departed from the ordinary laws, and
Iceland spar was almost the only crystal known to have the property in
question. At the present day, however, the substances which are known to
produce double refraction are far more numerous than those which do not
possess this property, for, by a more refined mode of examination than
the production of double images, Arago has been able to infer the
existence of a similar effect on light in a vast number of bodies.
Crystals have also been found which split up a ray of light entering
them into two rays, neither of which obeys the laws of Descartes. It
may, in fact, be said that, with the exception of water, and most other
liquids, of gelatine and other colloidal substances, and of
well-annealed glass, there are few bodies which do not exercise similar
power on light.
[Illustration:
FIG. 207.
]
On examining the two rays which emerge from a rhomb of Iceland spar, on
which only one ray of ordinary light has been allowed to fall, we find
that these emergent rays have acquired new and striking properties, of
which the incident ray afforded no trace; for, if we allow the two rays
emerging from a rhomb of the spar to fall upon a second rhomb, we shall
find, on viewing the images produced, that their intensity varies with
the position into which its second crystal is turned. Thus, if we place
a rhomb of the spar upon a dot made on a sheet of white paper, we shall
have, as already pointed out, two images of equal darkness. But, in
placing a second rhomb of the spar upon the first, in such a manner that
their _principal sections_ coincide, and the faces of one rhomb are also
parallel to the faces of the other, we shall still see _two_ equally
intense images of the dot, only the images will be more widely separated
than before, and no difference will be produced by separating the
crystals if the parallelism of the planes of their respective principal
sections be preserved. Here, then, is at once a notable difference
between a ray of ordinary light and one that emerges from a rhomb of
Iceland spar; for, in the case of rays of ordinary light, we have seen
that the second rhomb would divide each ray into two, whereas it is
incapable (in the position of crystals under consideration) of dividing
either the ordinary or the extraordinary ray which emerges from the
first rhomb. If, still keeping the second rhomb above the other, we make
the former rotate in a horizontal plane, we may observe that, as we turn
the upper crystal so that the planes of the _principal sections_ form a
small angle with each, each image will be doubled, and, as the upper
crystal is turned, each pair of images exhibits a varying difference of
intensity. The ordinary ray in entering the second crystal is divided by
it into a second ordinary ray and a second extraordinary ray, the
intensities of which vary according to the angle between the principal
sections. When the two principal sections are parallel to one plane,
that is, when the angle between them is either 0° or 180°, the
extraordinary image disappears, and only the ordinary one is seen, and
with its greatest intensity. When the two _principal sections_ are
perpendicular to each other, that is, when the second crystal has been
turned through either 90° or 270°, the extraordinary has, on the
contrary, its greatest intensity, and the ordinary one disappears. When
the principal section of the second crystal has been turned into any
intermediate position, such as through 45° and 135°, or any odd multiple
of 45°, both images are visible and have equal intensities. This
experiment shows that the two rays which emerge from the first crystal
have acquired new properties, that each is affected differently by the
second crystal, according as the crystal is presented to it in different
directions round the ray as an axis. The ray of light is no longer
uniform in its properties all round, but appears to have acquired
different sides, as it were, in passing through the rhomb of Iceland
spar. This condition is indicated by saying that the ray is _polarized_,
and the first rhomb of spar is termed the _polarizer_, while the second
rhomb, by which we recognize the fact that both the _ordinary_ and the
_extraordinary_ rays emerge having different sides, has received the
name of _analyser_. But, in order to study conveniently all the
phenomena in Iceland spar, we should have crystals of a considerable
size, otherwise the two rays do not become sufficiently separated so as
to make it an easy matter to intercept one of them while we examine the
other. A very ingenious mode of getting rid of one of the rays was
devised by Nicol, and as his apparatus is much used for experiments on
polarized light, we shall state the mode of constructing _Nicol’s
Prism_. It is made from a rhomb of Iceland spar, Fig. 207, in which _a_
and _b_ are the corners where the three obtuse angles meet, all equal.
If we draw through _a_ and _b_ lines bisecting the angles _d a c_ and _f
h g_, and join _a b_, these lines will all be in one plane, which is a
principal section of the crystal, and contains the axis, _a b_. Now
suppose another plane, passing through _a b_, to be turned so that it is
at right angles to the plane containing _a b_ and the bisectors: this
plane would cut the sides of the crystal in the lines _a i_, _i h_, _b
k_, _k a_; and in making the Nicol prism, the crystal is cut into two
along this plane, and the two pieces are then cemented together by
_Canada balsam_. A ray of light, R, entering the prism, undergoes double
refraction; but the ordinary ray, meeting the surface of the Canada
balsam at a certain angle greater than the limiting angle, is totally
reflected, and passes out of the crystal at O; while the extraordinary
ray, meeting the layer of balsam at a less angle than _its_ limiting
angle, does not undergo total reflection, but passes through the balsam,
and emerges in the direction of E, completely polarized, so that the ray
is unable to penetrate another Nicol’s prism of which the principal
section is placed at right angles to that of the first.
[Illustration:
FIG. 208.
]
Among other crystals which possess the property of doubly refracting,
and therefore of polarizing, is the mineral called _tourmaline_, which
is a semi-transparent substance, different specimens having different
tints. In Fig. 208, A, B, represent the prismatic crystals of
tourmaline, and C shows a crystal which has been cut, by means of a
lapidary’s wheel, into four pieces, the planes of division being
parallel to the axis of the prism. The two inner portions form slices,
having a uniform thickness of about 1/20 in., and when the faces of
these have been polished, the plates form a convenient polarizer and
analyser. Let us imagine one of the plates placed perpendicularly
between the eye and a lighted candle. The light will be seen distinctly
through it, partaking, however, of the colour of the tourmaline; and if
the plate be turned round so that the direction of the axis of the
crystal takes all possible positions with regard to the horizon, while
the plane of the plate is always perpendicular to the line between the
eye and the candle, _no change whatever will be seen in the appearance
of the flame_. But if we fix the plate of crystal in a given position,
let us say with the axial direction vertical, and place between it and
the eye the second plate of tourmaline, the appearances become very
curious indeed, and _the candle is visible or invisible according to the
position of this second plate_. When the axis of the second is, like
that of the first, vertical, the candle is distinctly seen; but when the
axis of the second plate is horizontal, no rays from the candle can
reach the eye. If the second plate be slowly turned in its own plane,
the candle becomes visible or invisible at each quarter of a revolution,
the image passing through all degrees of brightness. Thus the luminous
rays which pass through the first plate are polarized like those which
emerge from a crystal of Iceland spar. It is not necessary that the
plates used should be cut from the same crystal of tourmaline, for any
two plates will answer equally well which have been cut parallel to the
axes of the crystals which furnished them. In the case of tourmaline the
extraordinary ray possesses the power of penetrating the substance of
the crystal much more freely than the ordinary ray, which a small
thickness suffices to absorb altogether. It may be noted that in the
simple experiment we have just described, the plate of tourmaline next
the candle forms the _polarizer_, and that next the eye the _analyser_;
and that until the latter was employed, the eye was quite incapable of
detecting the change which the light had undergone in passing through
the first plate, for the unassisted eye had no means of recognizing that
the rays emerged with sides. The usual manner of examining light, to
find whether it is polarized, is to look through a plate of tourmaline
or a Nicol’s prism, and observe whether any change in brightness takes
place as the prism or plate is rotated. Now, it so happened that in 1808
a very eminent French man of science, named Malus, was looking through a
crystal of Iceland spar, and seeing in the glass panes of the windows of
the Luxembourg Palace, which was opposite his house, the image of the
setting sun, he turned the crystal towards the windows, and instead of
the two bright images he expected to see, he perceived only one; and on
turning the crystal a quarter of a revolution, this one vanished as the
other image appeared. It was, indeed, by a careful analysis of this
phenomenon that Malus founded a new branch of science, namely, that
which treats of polarized light; and his views soon led to other
discoveries, which, with their theoretical investigations, constitute
one of the most interesting departments of optical science, as
remarkable for the grasp it gives of the theory of light as for the
number of practical applications to which it has led.
The accidental observation of Malus led to the discovery that when a ray
of ordinary light falls obliquely on a mirror—not of metal, but of any
other polished surface, such as glass, wood, ivory, marble, or
leather—it acquires by reflection at the surface the same properties
that it would acquire by passing through a Nicol’s prism or a plate of
tourmaline: in a word, it is polarized. Thus, if a ray of light is
allowed to fall upon a mirror of black glass at an angle of incidence of
54° 35´, the reflected ray will be found to be polarized in the plane of
reflection—that is, it will pass freely through a Nicol’s prism when the
principal section is parallel to the plane of reflection; but when it is
at right angles to the latter, the reflected ray will be completely
extinguished by the prism—that is, it is completely polarized. If the
angle of the incident ray is different from 54° 35´, then the reflected
ray is not completely intercepted by the prism—it is not completely but
only partially polarized. The angle at which maximum polarization takes
place varies with the reflecting substance; thus, for water it is 53°,
for diamond 68°, for air 45°. A simple law was discovered by Sir David
Brewster by which the polarizing angle of every substance is connected
with its refractive index, so that when one is known, the other may be
deduced. It may be expressed by saying that the polarizing angle is that
angle of incidence which makes the reflected and the refracted rays
perpendicular to each other. The refracted rays are also found to be
polarized in a plane perpendicular to that of reflection.
[Illustration:
FIG. 209.—_Polariscope._
]
[Illustration:
FIG. 210.
]
[Illustration:
FIG. 211.—_Iceland Spar showing Double Refraction._
]