Similarly for n=2, n=3 there are two other spirals, whose positions can be obtained by turning the spiral for which n = = 0 backwards through two right angles, and three right angles respectively. When n = 4, the original spiral is reproduced. The forms of these spirals, which after their discoverer are usually known as Airy's Spirals, are shown in the figure on the last page, and also in figures 7 and 8 of the plate. At a considerable distance from the centre, a faint black cross makes its appearance, whose four arms are parallel and perpendicular to the plane of polarization of the incident light; also the spirals disappear and are replaced by circular rings. Now at a distance from the centre, k is nearly equal to zero; whence the intensity becomes The first equation gives the circular rings, whilst the latter equation gives the brushes. CHAPTER X. FRESNEL'S THEORY OF REFLECTION AND REFRACTION. 164. WHEN common light is incident upon the surface of a transparent medium, such as glass, it can be proved experimentally, that the proportion of the incident light which is reflected or refracted, depends upon the angle of incidence; and that the amount of light reflected is greater when the angle of incidence is large, than when it is small. It is also known, that when light proceeding from a denser medium, such as glass, is incident upon a rarer medium, such as air, at an angle greater than the critical angle, the intensity of the reflected light is very nearly equal to that of the incident light, and the reflection is said to be total. When the incident light is polarized in the plane of incidence, the effect produced by a reflecting medium is not very different from that produced upon common light; but when the light is polarized perpendicularly to the plane of incidence, it is found that the intensity gradually diminishes from grazing incidence, and very nearly vanishes, when the angle of incidence is equal to tan-1μ, where is the index of refraction of the reflecting substance; as the angle of incidence still further increases, the intensity of the reflected light increases to normal incidence. μ 165. That the intensity of light polarized perpendicularly to the plane of incidence is zero for a certain angle of incidence, was first discovered by Malus, who while examining with a prism of Iceland spar the light reflected from one of the windows of the Luxembourg palace at Paris, observed that for a certain position of the prism, one of the two images of the sun disappeared. On turning the prism round the line of sight, this image reappeared; and when the prism was turned through 90°, the second image BREWSTER'S LAW. 177 reappeared. More accurate experiments were afterwards made by Brewster', who discovered that when the reflector is an isotropic transparent substance, and the incident light is polarized perpendicularly to the plane of incidence, the intensity of the reflected light is zero, or very nearly so, when the angle of incidence is equal to tanμ. This discovery is known as Brewster's law, and the angle tan-1μ is called the polarizing angle. -1 μ 166. Brewster's law has been tested by Sir John Conroy for transparent bodies in contact with media other than air, in the following manner. A glass prism was placed in contact with water and with carbon tetrachloride respectively, and the polarizing angles were determined. Their values, as found by experiment, were as follows: The polarizing angles were then determined experimentally for water and carbon tetrachloride in contact with air, and the values of the polarizing angles for glass in contact with these substances were then calculated. The results were as follows: Polarizing angle in air observed 57° 14' calculated from observations in water 57° 28' in carbon tetrachloride 57° 01'. These results show, that within the limits of experimental error, Brewster's law holds good for glass in contact with water and carbon tetrachloride, as well as air; and that in all probability, it is true for most transparent bodies. 167. Crystalline substances, such as Iceland spar, also possess a polarizing angle as well as a critical angle. In isotropic media, the critical angle is equal to sinu; but in doubly refracting media, the values of the polarizing and critical angles cannot be so simply expressed. 168. Metallic substances, such as polished silver, possess a quasi-polarizing angle, since there is a particular angle of incidence at which the intensity of light polarized perpendicularly to the plane of incidence is a minimum. 1 Phil. Trans. 1815, p. 125. See also Lord Rayleigh, "On Reflection from Liquid Surfaces in the Neighbourhood of the Polarizing Angle," Phil. Mag. Jan. 1892. 2 Proc. Roy. Soc. vol. xxxI. p. 487. B. O. 12 169. In order to explain these experimental facts, it is necessary to determine the intensities of the reflected and refracted lights. This was first effected by Fresnel; and although his theory is not rigorous, it will be desirable to give an account of it in the present Chapter. Other theories based upon speculations respecting the physical constitution of the ether, which are developed according to strict dynamical principles, will be considered in subsequent chapters; and it will be found that most of them give results, which are substantially in accordance with those obtained by Fresnel. 170. We shall first calculate the rate at which energy flows across the reflecting surface. Let the incident vibration be 2π and let us consider the energy contained within a small cylinder whose cross section is dS, and whose sides coincide with the direction of propagation. If T denote the amount of kinetic energy per wave-length Since this amount of kinetic energy flows across dS in time 7, the rate at which kinetic energy flows across dS is πAV pdS/XT. Let dS' be any oblique section of the cylinder, which makes an angle e with dS, then if we assume that the energy of the wave is half kinetic and half potential, the rate at which energy flows across dS' is 2πA2 V2p cos edS'/XT........ The mean energy for unit of volume is ...(1). 2π2A2 V2p/X2..... as has already been shown in § 10. .(2), 171. We are now prepared to consider the problem of reflection and refraction. Let the axis of a be normal to the reflecting surface, and let the axis of z be perpendicular to the plane of incidence. POLARIZATION IN THE PLANE OF INCIDENCE. 179 Let the incident light be polarized in the plane of incidence, Ꮖ Y then the incident, reflected and refracted waves may be taken to be the real parts of w, w', w1, where Since the reflected and refracted waves are forced vibrations produced and maintained by the incident wave, it follows that the periods of the three waves must be the same; whence 0 move Since the traces of all three waves on the plane x = together, it follows that the coefficients of y must be the same in all three waves, whence .(5). If i, i, r be the angles of incidence, reflection and refraction, 172. We now require two equations connecting the amplitudes of the reflected and refracted waves. In order to effect this, Fresnel assumed, (i) that the displacements at the surface of separation are the same in the two media, (ii) that the rate at which energy flows across this surface is continuous. |