# On the Light from the Sky, Its Polarization and Colour (1871)

On the Light from the Sky, Its Polarization and Colour  (1871)
by John Strutt, 3rd Baron Rayleigh

It is now, I believe, generally admitted that the light which we receive from the clear sky is due in one way or another to small suspended’ particles which divert the light from its regular course. On this point the experiments of Tyndall with precipitated clouds seem quite decisive. Whenever the particles of the foreign matter are sufficiently fine, the light emitted laterally is blue in colour, and, in a direction perpendicular to that of the incident beam, is completely polarized.

About the colour there is no prima facie difficulty; for as soon as the question is raised, it is seen that the standard of linear dimension, with reference to which the particles are called small, is the wave-length of light, and that a given set of particles would (on any conceivable view as to their mode of action) produce a continually increasing disturbance as we pass along the spectrum towards. the more refrangible end; and there seems no reason why the colour of the compound light thus scattered laterally should not agree with that of the sky.

On the other hand, the direction of polarization (perpendicular to the path of the primary light) seems to have been felt as a difficulty. ‘Tyndall says, “......the polarization of the beam by the incipient cloud has thus far proved itself to be absolutely independent of the polarizing-angle, The law of Brewster does not apply to matter in this condition; and it rests with the undulatory theory to explain why. Whenever the precipitated particles are sufficiently fine, no matter what the substance forming the particles may be, the direction of maximum polarization is at right angles to the illuminating beam, the polarizing angle for matter in this condition being invariably 45°. This I consider to be a point of capital importance with reference to the present question” *. As to the importance there will not be two opinions; but I venture to think that the difficulty is imaginary, and is caused mainly by misuse of the word reflection. Of course there is nothing in the etymology of reflection or refraction to forbid their application in this sense ; but the words have acquired technical meanings, and become associated with certain well-known laws called after them. Now a moment's consideration of the principles according to which reflection and refraction are explained in the wave theory is sufficient to show that they have no application unless the surface of the disturbing body is larger than many square wave-lengths; whereas the particles to which the sky is supposed to owe its illumination must be smaller than the wave-length, or else the explanation of the colour breaks down. The idea of polarization by reflection is therefore out of place; and that “the law of Brewster does not apply to matter in this condition ” (of extreme fineness) is only what might have been inferred from the principles of the wave theory.

Nor is there any difficulty in foreseeing what, according to the wave theory, the direction of polarization ought to be. Conceive a beam of plane-polarized light to move among a number of particles, all small compared with any of the wave-lengths, The foreign matter, if optically denser than air, may be supposed to load the aether so as to increase its inertia without altering its resistance to distortion, provided that we agree to neglect effects analogous to chromatic dispersion. If the particles were away, the wave would pass on unbroken and no light would be emitted laterally. Even with the particles retarding the motion of the aether, the same will be true if, to counterbalance the increased inertia, suitable forces are caused to act on the aether at all points where the inertia is altered. These forces have the same period and direction as the undisturbed luminous vibrations themselves. The light actually emitted laterally is thus the same as would be caused by forces exactly the opposite of these acting on the medium otherwise free from disturbance ; and it only remains to see what the effect of such forces would be.

On account of the smallness of the particles, the forces acting throughout the volume of any one are all of the same intensity and direction, and may be considered as a whole. The determination of the motion in the aether, due to the action of a periodic force at a given point, requires, of course, the aid of mathematical analysis; but very simple considerations will lead us to a conclusion on the particular point now under discussion. In the first place there is a complete symmetry round the direction of the force. The disturbance, consisting of transverse vibrations, is propagated outwards in alt directions from the centre; and in consequence of the symmetry, the direction of vibration in any ray lies in the plane containing the ray and the axis; that is to say, the direction of vibration in the scattered or diffracted ray makes with the direction of vibration in the incident or primary ray the least possible angle. The symmetry also requires that the intensity of the scattered light should vanish for the ray which would be propagated along the axis; for there is nothing to distinguish one direction transverse to the ray from another. We have now got what we want. Suppose, for distinctness of statement, that the primary ray is vertical, and that the plane of vibration is that of the meridian. The intensity of the light scattered by a small particle is constant, and a maximum for rays which lie in the vertical plane running east and west, while there is no scattered ray along the north and south line. If the primary ray is unpolarized, the light scattered north and south is entirely due to that component which vibrates east and west, and is therefore perfectly polarized, the direction of its vibration being also east and west. Similarly any other ray scattered horizontally is perfectly polarized, and the vibration is performed in the horizontal plane. In other directions the polarization becomes less and less complete as we approach the vertical, and in the vertical direction itself altogether disappears.

So far, then, as disturbance by very small particles is concerned, theory appears to be in complete accordance with the experiments of Tyndall and others. At the same time, if the above reasoning be valid, the question as to the direction of the vibrations in polarized light is decided in accordance with the view of Fresnel. Indeed the observation on the plane of polarization of the scattered light is virtually only another form of Professor Stokes's original test with the diffraction-grating. In its present shape, however, it is free from certain difficulties both of theory and experiment, which have led different physicists who have used the other method to contradictory conclusions. I confess I cannot see any room for doubt as to the result it leads to[1].

The argument used is apparently open to a serious objection, which I ought to notice. It seems to prove too much. For if one disturbing particle is unable to send out a scattered ray in the direction of original vibration, it would appear that no combination of them (such as a small body may be supposed to be) could do so, at least at such a distance that the body subtends only a small solid angle. Now we know that when light vibrating in the plane of incidence falls on a reflecting surface at an angle of 45°, light is sent out according to the law of ordinary reflection, whose direction of vibration is perpendicular to that in the incident ray. And not only is this so in experiment, but it has been proved by Green[2] to be a consequence of the very same view as to the nature of the difference between media of various refrangibilities as has been adopted in this paper. The apparent contradiction, however, is easily explained. It is true that the disturbance due to a foreign body of any size is the same as would be caused by forces acting through the space it fills in a direction parallel to that in which the primary light vibrates; but these forces must be supposed to act on the medium as it actually is—that is, with the variable density. Only on the supposition of complete uniformity would it follow that no ray could be emitted parallel to the line in which the forces act. When, however, the sphere of disturbance is small compared with the wave-length, the want of uniformity is of little account, and cannot alter the law regulating the intensity of the vibration propagated in different directions.

Having disposed of the polarization, let us now consider how the intensity of the scattered light varies from one part of the spectrum to another, still supposing that all the particles are many times smaller than the wave- length even of violet light. The whole question admits of analytical treatment; but before entering upon that, it may be worth while to show how the principal result may be anticipated from a consideration of the dimensions of the quantities concerned.

The object is to compare the intensities of the incident and scattered rays; for these will clearly be proportional. The number (${\displaystyle i}$) expressing the ratio of the two amplitudes is a function of the following quantities:—${\displaystyle T}$, the volume of the disturbing particle; ${\displaystyle r}$, the distance of the point under consideration from it; ${\displaystyle \lambda }$, the wave-length; ${\displaystyle b}$, the velocity of propagation of light; ${\displaystyle D}$ and ${\displaystyle D'}$, the original and altered densities: of which the first three depend only on space, the fourth on space and time, while the fifth and sixth introduce the consideration of mass. Other elements of the problem there are none, except mere numbers and angles, which do not depend on the fundamental measurements of space, time, and mass. Since the ratio ${\displaystyle i}$, whose expression we seek, is of no dimensions in mass, it follows at once that ${\displaystyle D}$ and ${\displaystyle D'}$ only occur under the form ${\displaystyle D:D'}$, which is a simple number and may therefore be omitted. It remains to find how ${\displaystyle i}$ varies with ${\displaystyle T,\,r,\,\lambda ,\,b.}$

Now, of these quantities, ${\displaystyle b}$ is the only one depending on time; and therefore, as ${\displaystyle i}$ is of no dimensions in time, ${\displaystyle b}$ cannot occur in its expression. We are left, then, with ${\displaystyle T,\,r,}$ and ${\displaystyle \lambda }$; and from what we know of the dynamics of the question, we may be sure that ${\displaystyle i}$ varies directly as ${\displaystyle T}$ and inversely as ${\displaystyle r}$, and must therefore be proportional to ${\displaystyle T/\lambda ^{2}r}$, ${\displaystyle T}$ being of three dimensions in space. In passing from one part of the spectrum to another ${\displaystyle \lambda }$ is the only quantity which varies, and we have the important law :—

When light is scattered by particles which are very small compared with any of the wave-lengths, the ratio of the amplitudes of the vibrations of the scattered and incident light varies inversely as the square of the wave-length, and the intensity of the lights themselves as the inverse fourth power.

I will now investigate the mathematical expression for the disturbance propagated in any direction from a small particle which a beam of light strikes.

Let the vibration corresponding to the incident light be expressed by ${\displaystyle A\cos(2\pi bt/\lambda )}$. The acceleration is

 ${\displaystyle -A\left({\frac {2\pi }{\lambda }}b\right)^{2}\cos {\frac {2\pi }{\lambda }}bt;}$

so that the force which would have to be applied to the parts where the density is ${\displaystyle D'}$, in order that the wave might pass on undisturbed, is, per unit of volume,

 ${\displaystyle -(D'-D)A\left({\frac {2\pi }{\lambda }}b\right)^{2}\cos {\frac {2\pi }{\lambda }}bt.}$

To obtain the total force which must be supposed to act over the space occupied by the particle, the factor ${\displaystyle T}$ must be introduced. The opposite of this conceived to act at ${\displaystyle O}$ (the position of the particle) gives the same disturbance in the medium as is actually caused by the presence of the particle. Suppose, now, that the ray is incident along ${\displaystyle OY}$, and that the direction of vibration makes an angle a with the axis of ${\displaystyle x}$, which is the line of the scattered ray under consideration—a supposition which involves no loss of generality, because of the symmetry which we have shown to exist round the line of action of the force. The question is now entirely reduced to the discovery of the disturbance produced in the aether by a given periodic force acting at a fixed point in it. In his valuable paper “On the Dynamical Theory of Diffraction”[3], Professor Stokes has given a complete investigation of this problem; and I might assume the result at once, The method there used is, however, for this particular purpose very indirect, and accordingly I have thought it advisable to give a comparatively short cut to the result, which will be found at the end of the present paper. It is proved that if the total force acting at ${\displaystyle O}$, in the manner supposed be ${\displaystyle F\cos(2\pi bt/\lambda )}$, the resulting disturbance in the ray propagated along ${\displaystyle OX}$ is

 ${\displaystyle \zeta ={\frac {F\sin \alpha }{4\pi b^{2}Dr}}\cos {\frac {2\pi }{\lambda }}(bt-r).}$
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1. I only mean that if light, as is generally supposed, consists of transversal vibrations similar to those which take place in an elastic solid, the vibration must be normal to the plane of polarization. There is unquestionably a formal analogy between the two sets of phenomena extending over a very wide range; but it is another thing to assert that the vibrations of light are really and truly-to-and-fro motions of a medium having mechanical properties (with reference to small vibrations) like those of ordinary solids. The fact that the theory of elastic solids led Green to Fresnel’s formulae for the reflection and refraction of polarized light seems amply sufficient to warrant its employment here, while the question whether the analogy is more than formal is still left open.
2. Camb. Phil, Trans. vol. VII. 1887.
3. Camb. Phil. Trans. vol. II. p. 1, 1849.