the square of the amplitude of the incident vibrations is equal to the
sum of the squares of the amplitudes of the reflected and refracted
vibrations.
Fresnel obtained his formulae by assuming that the optical difference of media is due to a change in the effective density of the ether, the elastic it being the same—an assumption inconsistent with his theory of double refraction—and was led to the result that the vibrations are perpendicular to the plane of polarization. Franz Neumann and James MacCullagh, starting from the opposite assumption of constant density and different elasticities, arrived at the same formulae for the intensities of the reflected light polarized in the principal azimuths, but in this case the vibrations must be regarded as parallel to the plane of polarization. The divergence 0 these views has led to a large number of experimental investigations, instituted with the idea of deciding between them. In the main such investigations have only an academic interest, as, whatever theory of light be adopted, we have to deal with two vectors that are parallel and perpendicular respectively to the plane of polarization. Thus certain experiments of Otto H. Wiener (Wied. Ann., 1890, xl. 203) show that chemical action is to be referred to the latter of these vectors, but whether Fresnel’s or Neumann’s hypothesis be correct is only to be decided when we know if it be the mean kinetic energy or the mean potential energy that determines chemical action Similarly on the electromagnetic theory the electric or the magnetic force will be perpendicular to the plane of polarization, according as chemical action depends upon the electric or the magnetic energy. Lord Rayleigh (Scientific Papers, i 104) has, however, shown that the polarization of the light from the sky can only be explained on the elastic solid theory by Fresnel’s hypothesis of a different density, and from the study of Hertzian oscillations, in which the direction of the electric vibrations can be a priori assigned, we learn that when these are in the plane of incidence there is no reflection at a certain angle, so that the electric force is perpendicular to the plane of polarization.
It has been supposed in the above that the medium into which the light enters at the reflecting-surface is the more refracting. In the contrary case, total reflection commences as soon as sin i＝μ−1, μ being still the relative refractive index of the more highly refracting medium; and for greater angles of incidence r becomes imaginary. Now Fresnel's formulae were obtained by assuming that the incident, reflected and ref racted vibrations are in the same or opposite phases at the interface of the media, and since there is no real factor that converts cos T into cos (T+ρ), he inferred that the occurrence of imaginary expressions for the coefficients of vibration denotes a change o phase other than vr, this being represented by a change of sign If this be so, it is clear that the factor √−1 denotes a change of phase of π/2, since this twice repeated converts cos T into cos (T+π)＝−cos T, and henoe that the factor a+b√−1 represents a change of phase of tan−1(b/a). Applying this interpretation to the formulae given above, it follows that when the incident light is polarized at an azimuth α to the plane of incidence and the second medium is the less refracting, the reflected light at angles of incidence exceeding the critical angle is elliptically polarized with a difference of phase Δ between the components polarized in the principal azimuths that is given by
tan (Δ/2)＝cot i√(1−μ−2 cosec2 i).
Thus Δ is zero at grazing incidence and at the critical angle, and attains its maximum value π−4 tan−1(1/μ) at an angle of incidence given by sin2 i=2/(2μ2+1).
It is of some interest to determine under what conditions it is possible to obtain a specified difference of phase Solving for cot2 'i we obtain
2 cot2i＝(μ2 ± √[(μ2−tan2 (π−Δ)/4) {μ2−cot2 (π−Δ)/4}],
and since tan {(π−Δ)/4} is less than unity, μ must exceed cot {(π−Δ)/4} if cot2i in to be real. Thus if Δ=π/2, μ must exceed π/8 or 2·414, that is, the substance must be at least as highly refracting as a diamond. if Δ = π/4, μ must be greater than 3π/16 or 1·4966, and when this is the case it is possible by two reflections to convert into a circularly polarized stream a beam of light polarized at 45° to the plane of incidence This is the principle of Fresnel’s rhomb, that is sometimes employed instead of a quarter-wave plate for obtaining a strc-am of circularly polarized light It consists of a parallelepiped glass so constructed that light falling normally on one end emerges at the other after two internal reflections at such an angle as to introduce a relative retardation of phase of π/4 between the components polarized in the principal azimuths.
Fresnel’s formulae are sufficiently accurate for most practical purposes, but that they are not an exact representation of the facts of reflection was shown by Sir David Brewster and bv Sir G. B. Airy. Detailed investigations by J. C. Jamin, G. H. Quincke, C. W. Wernicke and others have established that in general light polarized in any but the principal azimuths becomes elliptically polarized by reflection, the relative retardation of phase of the components polarized in these azimuths becoming π/2 at a certain angle of incidence, called the principal incidence. In some cases it is the component polarized in the plane of incidence that is most retarded and the reflection is then said to be positive in the case of negative reflection the reverse takes place. It was at first supposed that the defect of Fresnel’s formulae was due to the neglect of the superficial undulations that, on a rigorous elastic solid theory of the ether, are called into existence at reflection and refraction. But the result of taking these into account is far from being in accordance with the facts, and experiments of Lord Rayleigh and Paul Drude make it probable that we ought to assume that the transition from one medium to another, though taking place in a distance amounting to about one fiftieth of a wave-length, is gradual instead of abrupt. The effect of such a transition-layer can easily be calculated, at least approximately; but it is of little use to take account of it except in the case of a theory of reflection that gives Fresnel's formulae as the result of an abrupt transition. Lord Rayleigh has pointed out that all theories are defective in that they disregard the fact that one at least of the media is dispersive, and that it is probable that finite reflection would result at the interface of media of different dispersive powers, even in the case of waves for which the refractive indices are absolutely the same.
A more pronounced case of elliptic polarization by reflection is afforded by metals Formulae for metallic reflection may be obtained from Fresnel’s expressions by writing the ratio sin i/ sin r equal to a complex quantity, and interpret in the imaginary coefficients in the manner explained above. The optical constants (refractive index and co-efficient of extinction) of the metal may then be obtained from observations of the principal incidence and the elliptic polarization then produced. A detailed investigation of these constants has been made by Drude (Wied. Ann., 1890, xxxix. 504), who has found the remarkable result that copper, gold, magnesium and silver have refractive indices less than unity, and this has been completely confirmed by observations with metallic prisms of small refracting angle. He further showed that except in the cases of co per, lead and gold the dispersion is abnormal—the index for red light being greater than that for sodium light. The higher the co-efficient of extinction for light of a given period, the more copious will be reflection of that constituent of a mixed pencil. This fact has been employed for separating waves of large wavelength, and in this way Waves of length 0·061 mm. have been isolated by five successive reflections from the surface of sylvite.
The Study of Polarization.—The best method of obtaining a strong beam of polarized light is to isolate one of the streams into which a beam of common light is resolved by double refraction. This is effected in polarizing prisms of the earlier type, devised by A. M. de Rochon, H. H de Sénarmont and W H Wollaston, by blocking off one of the streams with a screen, sufficient lateral separation being obtained by combining two equal crystalline prisms cut differently with respect to the optic axis—an arrangement that achromatizes more or less completely the pencil that is allowed to pass. In a second type, called Nicol’s prisms, one stream is removed by total reflection. Theoretically the best construction for prisms of this class is the following a rectangular block of Iceland spar, of length about four times the width and having its end and two of its side faces parallel to the optic axis, is cut in half by a lane parallel to the optic axis and making an angle of about 14° with the sides; the two halves are then reunited with a cement whose refractive index is between the ordinary and extraordinary indices of the spar and as nearly as possible equal to the latter. Thus constructed, the prism produces no lateral shift of the transmitted pencil; a conical pencil, incident directly, has nearly constant polarization over its extent, and consequently the error in determining the polarization of a parallel pencil, incident not quite normally, is a minimum. In a Nicol’s prism it is the extraordinary stream that passes; in a prism suggested by E. Sang and sometimes called a Bertrand's prism, it is the ordinary stream that is utilized This is made by fixing a thin crystalline plate between two glass prisms turned in opposite directions by a cement of the same refractive index as the glass. This refractive index should be equal to the greatest index of the plate, and with a biaxal late the mean axis of optical symmetry should be parallel to its faces and in the normal Section of the prisms, while with an uniaxal plate the optic axis should be in a plane perpendicular to this normal section. These prisms have the advantage of economy of material and of a greater field than the ordinary Nicol’s prism, but a difficulty seems to be experienced in finding a suitable permanent cement.
For an accurate determination of the plane of polarization analysers that act by extinction are not of much practical use, and a different device has to be employed. Savart’s analyser consists of a Savart’s plate (see above) connected to a Nicol’s prism, the principal section of which bisects the angle between the principal planes of the plate: the plane of polarization is determined by turning the analyser until the bands, ordinarily seen, disappear, in which case it is parallel to one of the principal planes of the plate. Half-shade analysers depend upon the facility with which the eye can distinguish slight differences in the intensities of two streams seen in juxtaposition, when the illumination is not too bright. The field is divided into two parts that for most positions of the analyser have different intensities, and the setting is effected by turning the analyser until both halves are equally dark. These instruments are very sensitive, but care must be taken to avoid errors caused by changes in the relative intensities of parts of the source of light—a precaution that is sometimes overlooked in furnishing polarimeters with these analysers. In J. H. Jellet’s and M. A. Cornu’s analysers
