XVI. Measurements of the Polarization of the Light reflected by the Sky and by one or more Plates of Glass. By Professor EDWARD C. PICKERING, Boston, U.S.* THE HE following observations, which will be published in full in the 'Proceedings of the American Academy of Arts and Sciences,' were conducted to test Fresnel's formula for the reflection of light. He showed that, if the light was polarized in the plane of incidence, the amount reflected would be sin2 (i-r) A= while if polarized in a plane perpendicular to it sin2 (i+r)' the proportion would be B= tan2 (i-r) tan2 (1+r)' i and r representing the angles of incidence and refraction respectively. Natural light may be regarded as composed of two equal beams polarized at right angles, hence the amount reflected R= (A+B)= {/ sin2 (i-r) tan (i-r)` + a formula which may be applied to any special case by substituting proper values for i and r. The value of A evidently increases as i varies from 0° to 90°. That of B, on the other hand, diminishes from 0° until i+r=90°, when it equals 0; or at this angle, which is that of total polarization, all of the ray B is transmitted, all the reflected beam being polarized in the plane of incidence. When i=90°, A=1, B=1; hence all the light is 2 reflected. When i=0°, A, B, and R equal (1)2; n+ ; hence the reflected light increases with n, being zero when n=1, and 100 per cent. when n= ∞. Many familiar phenomena are thus readily accounted for-for instance, the brightness of the diamond, the covering power of white lead as a paint, and the brilliancy of wet or varnished stones and woods. A curious case presents itself when n=1+dn, or differs from unity only by an infinitesimal amount. A then becomes equal to 4dn2 (1 + tan2 )2, and B to dn2 (1 — tan2 i)2. When i=0, A, B, and R equal dn2; and this quantity is accordingly taken as the unit in Table I. The first column gives various values of the angle of incidence, the second and third the corresponding values of A and B, the fourth the amount of light reflected, and the fifth the degree of polarization. The other columns will be explained hereafter. This Table is evidently applicable to all cases where the media bounding the surface have nearly the same index of refraction, whether its absolute amount is great or small. *Communicated by the Author. TABLE I. Light reflected when n is near unity, or equals 1+dn. The most important application of Fresnel's formula is to the case of glass, where n somewhat exceeds 1.5. The first portion of Table II. gives in an abbreviated form the result of a computation for various values of i, A, B, R, and the polarization of the reflected and refracted rays. When, as frequently happens in the case of plates of glass, the ray passes through several parallel surfaces, a portion of the light reflected back by the second surface is again intercepted by the first surface. It may readily be proved that, if A is the amount reflected by a single surface, the amount transmitted including this internal reflection will be 1-A while if no internal reflection took place it would 1+(m-1)A' be only (1-A)". In Table II. the values of A, B, R and of the polarization are given for 2, 8, and 20 surfaces, corresponding to 1, 4, and 10 plates of glass. In all these cases the index n=1.55. the Light reflected by the Sky and by Plates of Glass. 129 TABLE II.-Light reflected by 1, 2, 8, and 20 surfaces. Phil. Mag. S. 4. Vol. 47. No. 310. Feb. 1874. K These results are perhaps better shown in figs. 1 and 2, in which abscissæ represent values of i, and ordinates percentages of polarization. În fig. 1 the four highest curves represent the polarization of the beams reflected by 1, 2, 8, and 20 surfaces. The other four curves give the corresponding refracted beams. Fig. 2 gives all the curves of Table II. relating to 20 surfacesthe five curves corresponding to A, B, the intensity of the refracted beam, and the polarization of both the reflected and refracted beams. When i=0, both the reflected and refracted beams are unpolarized. With 10 plates of glass about half the light is reflected, the transmitted ray being but little brighter than that reflected. With 1 or 2 surfaces the reflected beam increases as i increases; with 8 surfaces it remains nearly constant up to 50°; while with 20 surfaces a marked diminution is perceived. This very remarkable result may be expressed by saying that 10 plates of glass transmit more light obliquely than normally. The appearance to the eye confirms this result, but it deserves a careful photometric proof. At 57° the reflected ray is, of course, in all cases totally polarized; but at other angles the amount of polarization is greater the less the number of surfaces, instead of the contrary, as might have been anticipated. With the refracted ray quite a different law holds. For 1 surface the polarization increases from 0° to 90°; with 2 surfaces it becomes sensibly constant near 90°; while with a larger number a distinct maximum is obtained. It is commonly supposed that the greatest effect is obtained at the angle of total polarization. But the maximum is sensibly beyond this, unless a very large number of plates are employed; and hence it seems probable that a bundle of plates, polarizing by refraction, would give the best results if set at a greater angle than 57°, as 65° or 70°. The transmitted ray, however, diminishes rapidly for large angles of incidence. A very large number of plates are required to render the polarization nearly complete, which accounts for the light always remaining when even the best polariscopes by refraction are crossed. At 90° all the refracted beams are polarized by the same amount of 412 per cent. Or, at grazing incidence, the amount of polarization is independent of the number of plates, one polarizing as completely as a hundred. This number, 41.2, may be obtained as follows. Differentiate the value of A in terms of i and r, and make i=90°, when the refracted beam will equal 1-A = 4 tan r di = 3.376 di, since when i = 90° r=40° 10'7. In the same way 1-B= di=8·115 di; and applying to them the formula for the polarization of the refracted beam, we find it equal to 41.2. 8 sin 2r To show how far these effects are due to internal reflection, another Table was computed by the formula (1-a)", or supposing that no internal reflection took place. A comparison showed that while the reflected beam is affected but little, a great change takes place in the transmitted light. The results are shown by the dotted lines in figs. 7, 8, and 10, and will be discussed below. To test the above conclusions, two experimental methods may be employed. First, by means of a photometer, to determine the amount of light in any given case; and, secondly, by means of a polarimeter, to determine the percentage of polarization of the reflected and refracted rays. The latter method has been employed in the following experiments. The instrument commonly used to measure the amount of polarization was invented by Arago, and is called a polarimeter. It consists of a Nicol's prism and Savart's plates, in front of which are several glass plates, free to turn, and carrying an index which moves over a graduated circle, thus showing the angle through which they have been rotated. The prism and plates form a Savart's polariscope, which gives coloured bands with either light or dark centre, according as the plane of the prism is parallel or perpendicular to the plane of polarization. When the plates are so placed that the light passes through them normally, they have no effect on it; but when turned, they polarize it in a plane parallel to the axis of rotation, and by an amount dependent on the angle. Let the instrument be so set that the axis of rotation shall be perpendiular to the plane of polarization, and the plates set at zero. The bands will then be visible, the centre one being bright. As the plates are turned the bands become fainter, until the polarization neutralizes that originally present in the beam; beyond this point the bands reappear dark-centred. The amount of polarization is thus readily determined by turning the plates until the bands disappear, when the angle is reduced to percentages by means of a Table. The difficulty of computing this Table, however, is the real objection to the use of this instrument. It may be determined by the formulæ given in the first part of this paper; but it, of course, then fails to prove them. Moreover no account is taken of imperfect transparency, dust on the surface, and other sources of error. An excellent way of forming this Table experimentally is to view through the instrument a beam of light totally polarized. If now the plane of polarization of the beam is changed, the percentage of polarization will alter, being zero when it is inclined 45° to the axis of the plates, and wholly polarized at an angle of 0° or 90°. At any angle a the beam may be regarded as composed of two, cos a polarized vertically, and sin a polarized |