Part 48

“The swan, on still St. Mary’s Lake, Floats double, swan and shadow.”

The landscape painter is always gratified if he can introduce into his picture some piece of water, and it can hardly be doubted that much of the charm of lakes and rivers is due to their power of reflecting. Look on Fig. 201, a view of some buildings at Venice; and, in order to see how much of its beauty is owing to the quivering reflections, imagine the impression it would produce were the place of the water occupied by asphalte pavement, or a grass lawn. The condition of the reflections here represented is perhaps even more pleasing than that produced by perfect repose: they are in movement, and yet not broken and confused:

“In bright uncertainty they lie, Like future joys to Fancy’s eye.”

_REFRACTION._

That light moves in straight lines is a statement which is true only 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.

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.

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.

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.