Determination of the Thickness and the Orientation of a Calcite Plate by Interference Fringes.

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Megan Barnett
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1 Determination of the Thickness and the Orientation of a Calcite Plate by Interference Fringes. S. T. NAKAMURA AND R. TADIME. [READ SEPTEMBER 19, 1914.] For the explanation of the interference figures by double refraction in a crystal, it is customary to ignore the difference in the angles of refraction of the two refracted waves, and form an approximate formula (1) for the phase difference between them. Such formula serves fairly well to explain the general features of the phenomena, but quantitatively it fails to give correct dimensions of the fringes, except those of lower orders. During our third year's course in the experimental physics in the Imperial University of Tokyo, we made a series of observations on a plate of calcite cut nearly perpendicular to the optical axis, and measured the dimensions of fringes up to the one-hundredth order. By comparing them with those calculated by a formula obtained by us, it was found that they agreed very well with each other. It may be allowed us to present our result to the Society, as it affords an example of a method of measuring the thickness (2) and the orientation of a crystal plate by means of interference fringes. The calcite plate, obtained from Dr. Steeg and Reuter in Homburg, Germany, was secured in cork, and had a thickness of about 1mm. It was detached from its cork support and examined for the evenness of the faces by Newton's rings by mounting it on a Fizeau-Abbe's dilatometer of Zeiss, and for their parallelism by measuring its thickness by an Abbe's thickness-meter also of Zeiss. It was found that the plate is not exactly plane-parallel over its whole area. Those parts of the plate unfit for the observation were therefore covered by Chinese ink and screened off, and only an area of a few square millimetres with an irregular boundary was used. The mean thickness of the plate as determined by the thickness-meter is 1 E068mm. (1) See e. g. Drude, Lehrb. d. Optik (3 Aufl., Leipzig), p (2) M. Kawamura, Mem. of College of Eng., Kyushu Imp. Univ., [1] 1, 1913, p.

The plate was mounted on a polarization-spectrometer made by Fuess. The horizontal circle of the instrument has two verniers by which we can read to half a minute.

2 The plate was mounted on a polarization-spectrometer made by Fuess. The horizontal circle of the instrument has two verniers by which we can read to half a minute. The observing telescope has a Gauss's ocular, by which the plate was adjusted to have its normal exactly parallel to the plane of the horizontal circle. The slit of the collimator was illuminated by the green light of mercury, for which a glass-mercury lamp and a Wulfing's monochromator were used, With crossed Nicols, whose planes of polarization were inclined 45 to the horizon, it was seen at once that the normal of the plate does not coincide with the optical axis. The plate was therefore turned in its own plane till its principal section was exactly horizontal. The apparent inclination, i. e. in air, of the optical axis to the plate-normal was found to be about 8'. As the traces of the principal isogyres were pretty broad, their intersection could not be exactly determined, so that the last number gives only an approximate value of the inclination. With the plate thus adjusted, and the axis of collimation of the telescope in one line with that of the collimator, the plate was turned about the vertical axis of the instrument, and the positions of every fifth interference rings on both sides of the optical axis were read off on the circle As the position of the plate-normal could be accurately determined by the Gaussian ocular, we thus got angles of incidence corresponding to these fringes. If a plane wave of polarized light be incident on a crystalline plate, with its wave normal lying in the principal section of the plate, and its angle of incidence equal to i, and the angles of refraction equal to r1 and r2, then the phase difference ƒâ between the two refracted waves at their emergence from the plate is given by where d is the thickness of the plate, and ƒé the wave length in vacuum of the light used, and further ƒê1 and ƒê2 the indices of refraction of the two waves along their respective wave normals. Considering a uniaxial crystal with its optical axis inclined to the plate-normal by an angle ƒæ, if ƒö and ƒã be the two principal indices of refraction, then for an ordinary wave and for an extraordinary

3 It can be easily shown that r2 satisfies the following equation and hence ƒâ in any case can be calculated. If we take the particular case in which ƒæ=0, then

4 and again if we take the case in which ƒæ is small as in our present case (3) neglecting those terms involving small quantities of orders higher than the second in ƒæ. The sign of ƒæ in the last equation is to be taken positive when the wave normal and the optical axis lie on the same side of the plate-normal, and negative in the contrary case. An inter ference fringe of the mth order is observed when Hence the angle of incidence for the wave producing the fringe of the mth order is given by (4) Let i for the two sides of the plate-normal be ir and il, then and Subtracting and adding these two equations, and putting we get where terms involving small quantities ƒæ and ƒ of orders higher than the second are neglected. These two equations are used for determining the inclination ƒæ of the optical axis and the thickness d of the plate from the observed values of i. The results of the calculation are given in the accompanying table. The wave length ƒé of the green mercury line is taken to be 0 E5461ƒÊ, and the indices of refraction of calcite for that light at the room temperature Our final result is

5 so that the apparent inclination of the optical axis to the plate-normal ought to have been ƒöƒæ=11'0 instead of 8' as before stated. Using this value of d, we calculated i0 by the formula (6). These values of i0, given in the last column of the table, show their close agreement with the observed values. We take this opportunity of ex pressing our cordial thanks to Prof. Nakamura for his suggesting the subject and for his kind guidance during the experimen

THE DIFFRACTION GRATING SPECTROMETER

Purpose Theory THE DIFFRACTION GRATING SPECTROMETER a. To study diffraction of light using a diffraction grating spectrometer b. To measure the wavelengths of certain lines in the spectrum of the mercury