consider the section σ of the pencil by some intermediate plane, and a bundle of rays starting from the points of σ1 and reaching those of σ2 after having all passed through a point of that section σ.
(c) If in the last theorem the system of bodies is symmetrical around the straight line AB, we can take for σ1 and σ2 circular planes having AB as axis. Let h1 and h2 be the radii of these circles, i.e. the linear dimensions of an object and its image, ε1 and ε2 the infinitely small angles which a ray R going from A to B makes with the axis at these points. Then the above formula gives μ1h1ε1 = μ2h2ε2, a relation that was proved, for the particular case μ1 = μ2 by Huygens and Lagrange. It is still more valuable if one distinguishes by the algebraic sign of h2 whether the image is direct or inverted, and by that of ε2 whether the ray R on leaving A and on reaching B lies on opposite sides of the axis or on the same side.
The above theorems are of much service in the theory of optical instruments and in the general theory of radiation.
11. Phenomena of Interference and Diffraction.—The impulses or motions which a luminous body sends forth through the universal medium or aether, were considered by Huygens as being without any regular succession; he neither speaks of vibrations, nor of the physical cause of the colours. The idea that monochromatic light consists of a succession of simple harmonic vibrations like those represented by the equation (5), and that the sensation of colour depends on the frequency, is due to Thomas Young[1] and Fresnel,[2] who explained the phenomena of interference on this assumption combined with the principle of super-position. In doing so they were also enabled to determine the wave-length, ranging from 0.000076 cm. at the red end of the spectrum to 0.000039 cm. for the extreme violet and, by means of the formula (6), the number of vibrations per second. Later investigations have shown that the infra-red rays as well as the ultra-violet ones are of the same physical nature as the luminous rays, differing from these only by the greater or smaller length of their waves. The wave-length amounts to 0.006 cm. for the least refrangible infra-red, and is as small as 0.00001 cm. for the extreme ultra-violet.
Another important part of Fresnel’s work is his treatment of diffraction on the basis of Huygens’s principle. If, for example, light falls on a screen with a narrow slit, each point of the slit is regarded as a new centre of vibration, and the intensity at any point behind the screen is found by compounding with each other the disturbances coming from all these points, due account being taken of the phases with which they come together (see Diffraction; Interference).
12. Results of Later Mathematical Theory.—Though the theory of diffraction developed by Fresnel, and by other physicists who worked on the same lines, shows a most beautiful agreement with observed facts, yet its foundation, Huygens’s principle, cannot, in its original elementary form, be deemed quite satisfactory. The general validity of the results has, however, been confirmed by the researches of those mathematicians (Siméon Denis Poisson, Augustin Louis Cauchy, Sir G. G. Stokes, Gustav Robert Kirchhoff) who investigated the propagation of vibrations in a more rigorous manner. Kirchhoff[3] showed that the disturbance at any point of the aether inside a closed surface which contains no ponderable matter can be represented as made up of a large number of parts, each of which depends upon the state of things at one point of the surface. This result, the modern form of Huygens’s principle, can be extended to a system of bodies of any kind, the only restriction being that the source of light be not surrounded by the surface. Certain causes capable of producing vibrations can be imagined to be distributed all over this latter, in such a way that the disturbances to which they give rise in the enclosed space are exactly those which are brought about by the real source of light.[4] Another interesting result that has been verified by experiment is that, whenever rays of light pass through a focus, the phase undergoes a change of half a period. It must be added that the results alluded to in the above, though generally presented in the terms of some particular form of the wave theory, often apply to other forms as well.
13. Rays of Light.—In working out the theory of diffraction it is possible to state exactly in what sense light may be said to travel in straight lines. Behind an opening whose width is very large in comparison with the wave-length the limits between the illuminated and the dark parts of space are approximately determined by rays passing along the borders.
This conclusion can also be arrived at by a mode of reasoning that is independent of the theory of diffraction.[5] If linear differential equations admit a solution of the form (5) with A constant, they can also be satisfied by making A a function of the coordinates, such that, in a wave-front, it changes very little over a distance equal to the wave-length λ, and that it is constant along each line conjugate with the wave-fronts. In cases of this kind the disturbance may truly be said to travel along lines of the said direction, and an observer who is unable to discern lengths of the order of λ, and who uses an opening of much larger dimensions, may very well have the impression of a cylindrical beam with a sharp boundary.
A similar result is found for curved waves. If the additional restriction is made that their radii of curvature be very much larger than the wave-length, Huygens’s construction may confidently be employed. The amplitudes all along a ray are determined by, and proportional to, the amplitude at one of its points.
14. Polarized Light.—As the theorems used in the explanation of interference and diffraction are true for all kinds of vibratory motions, these phenomena can give us no clue to the special kind of vibrations in light-waves. Further information, however, may be drawn from experiments on plane polarized light. The properties of a beam of this kind are completely known when the position of a certain plane passing through the direction of the rays, and in which the beam is said to be polarized, is given. “This plane of polarization,” as it is called, coincides with the plane of incidence in those cases where the light has been polarized by reflection on a glass surface under an angle of incidence whose tangent is equal to the index of refraction (Brewster’s law).
The researches of Fresnel and Arago left no doubt as to the direction of the vibrations in polarized light with respect to that of the rays themselves. In isotropic bodies at least, the vibrations are exactly transverse, i.e. perpendicular to the rays, either in the plane of polarization or at right angles to it. The first part of this statement also applies to unpolarized light, as this can always be dissolved into polarized components.
Much experimental work has been done on the production of polarized rays by double refraction and on the reflection of polarized light, either by isotropic or by anisotropic transparent bodies, the object of inquiry being in the latter case to determine the position of the plane of polarization of the reflected rays and their intensity.
In this way a large amount of evidence has been gathered by which it has been possible to test different theories concerning the nature of light and that of the medium through which it is propagated. A common feature of nearly all these theories is that the aether is supposed to exist not only in spaces void of matter, but also in the interior of ponderable bodies.
15. Fresnel’s Theory.—Fresnel and his immediate successors assimilated the aether to an elastic solid, so that the velocity of propagation of transverse vibrations could be determined by the formula v = √(K/ρ), where K denotes the modulus of rigidity and ρ the density. According to this equation the different properties of various isotropic transparent bodies may arise from different values of K, of ρ, or of both. It has, however, been found that if both K and ρ are supposed to change from one substance to another, it is impossible to obtain the right reflection formulae. Assuming the constancy of K Fresnel was led to equations which agreed with the observed properties of the reflected light, if he made the further assumption (to be mentioned in what follows as “Fresnel’s assumption”) that the vibrations of plane polarized light are perpendicular to the plane of polarization.
- ↑ Phil. Trans. (1802), part i. p. 12.
- ↑ Œuvres complètes de Fresnel (Paris, 1866). (The researches were published between 1815 and 1827.)
- ↑ Ann. Phys. Chem. (1883), 18, p. 663.
- ↑ H. A. Lorentz, Zittingsversl. Akad. v. Wet. Amsterdam, 4 (1896), p. 176.
- ↑ H. A. Lorentz, Abhandlungen über theoretische Physik, 1 (1907), p. 415.
