§ 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.
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).
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).
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.
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.