Abstract

<p>Reflection and refraction of light at internal and external faces of birefringent crystals are discussed using a method devised by Max Planck for treating a particular case. General cases where a plane-polarized incident beam gives rise to (i) two plane-polarized refracted beams and a single plane-polarized reflected beam at external face and (ii) one plane-polarized refracted beam and two plane-polarized reflected beams at internal face are studied. Every beam, whatever the azimuth of its plane of polarization, is considered as a whole, unlike the usual practice of splitting a beam into two components, one polarized in the plane of incidence and the other polarized in a plane perpendicular to the latter.</p><p>Study shows that when a plane-polarized incident beam gives rise to a single plane-polarized reflected beam and a single plane-polarized refracted beam, the magnetic fields of the three beams are coplanar and the plane of the magnetic fields make an angle Z⌃<sub>b</sub> = tan<sup>-1</sup> | csc (Â-B⌃) · csc (B⌃-Ĉ) · sin<sup>2</sup>B⌃ · tan b⌃ |, with the plane of polarization of the refracted beam, where Â, B⌃, and Ĉ are, respectively, the angles of incidence refraction, and reflection and b⌃ is the small angle between the refracted beam and its ray.</p><p>The main conclusions of the paper were verified with a cleaved rhomb of calcite using a spectrometer fitted with a polarizer and analyzer. The azimuths of the planes of polarization (i) α⌃<sub>w</sub> and α⌃<sub>t</sub> of the incident beam, (ii) γ⌃<sub>w</sub> and γ⌃<sub>t</sub> of the reflected beam, and (iii) β⌃<sub>w</sub> and β⌃<sub>t</sub>, of the beams emerging through the rhomb, when there is only a single beam in the refracted light at the first surface are determined for a full range from +90° to -90° of the angle of incidence and for a fixed orientation of the rhomb.</p><p>The variation of γ⌃, the azimuth of the plane of polarization of reflected beam, with the variation of α⌃, the azimuth of the plane of polarization of the incident beam at the first face of the rhomb is also determined. The coplanar law of the magnetic fields has also been verified for a case where both the incident and reflected beams were extra-ordinary beams inside calcite. Experimental results confirm all the theoretical conclusions. To aid the visualization of the variation of azimuths, graphs are drawn on spheres.</p>

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  1. The incident beam is always named Å, the refracted beam is named B˙, and the reflected beam is named C˙.
  2. The subscript s indicates that the beam in the anisotropic medium is the slow beam for the given direction. The subscript attached to the associated beam, incident, reflected or refracted in an isotropic medium is w. Similarly, for the faster beam in an anisotropic medium the subscript added is ƒ and the subscript t is added to the associated beam in the isotropic medium. When the incident beam is a slow beam in an anisotropic medium and it gives rise to a slow reflected beam, the subscript added is ss and when it gives rise to a fast reflected beam the subscript added is sƒ. In the same manner, the combinations of ƒs and ƒƒ are used. The same procedure is followed for the subscripts of directions, angles, magnetic fields, and electric fields. The extra subscript r is added for those connected with a ray.
  3. There are two different circumstances in which one of the internally reflected beams may be absent. In one case, the azimuths of the planes of polarization of the incident and reflected beams are such that all the reflected energy appears in one of the reflected beams and thus the intensity of illumination of the second reflected beam is zero. Such cases are treated in this paper. For a second case where one of the reflected beams is absent owing to "total refraction" of one component of the incident beam and where the corresponding refracted beam is elliptically polarized, see, V. C. Varghese, J. Opt. Soc. Am. 57, 90 (1967).
  4. Positive normal to a given face and for a given incidence is the direction of the incident beam when the corresponding reflected beam retraces the path of the incident beam. Thus, the direction of the positive normal depends on the side of the surface on which light is incident. All angles of incidence, refraction, and reflection are measured in the positive counterclockwise direction from the same normal.
  5. For experimental confirmation that the trajectory of the energy of light is the direction of the ray and not the direction of the beam, see, V. C. Varghese, J. Opt. Soc. Am. 57, 86 (1967).
  6. In Eq. (7d), when 90°>Â>0°, both Ĉ and Ĉƒƒ, are between 90° and 180°. When 360°>Â>270°, Ĉ, and Ĉƒƒ are between 180° and 270°.
  7. M. Planck, Theory of Light (Macmillan and Co., London, 1932), Ip. 165.
  8. I. Todhunter and J. G. Leathem, Spherical Trigonometry Macmillan and Co., London, 1949), p. 111.

Leathem, J. G.

I. Todhunter and J. G. Leathem, Spherical Trigonometry Macmillan and Co., London, 1949), p. 111.

Planck, M.

M. Planck, Theory of Light (Macmillan and Co., London, 1932), Ip. 165.

Todhunter, I.

I. Todhunter and J. G. Leathem, Spherical Trigonometry Macmillan and Co., London, 1949), p. 111.

Varghese, V. C.

There are two different circumstances in which one of the internally reflected beams may be absent. In one case, the azimuths of the planes of polarization of the incident and reflected beams are such that all the reflected energy appears in one of the reflected beams and thus the intensity of illumination of the second reflected beam is zero. Such cases are treated in this paper. For a second case where one of the reflected beams is absent owing to "total refraction" of one component of the incident beam and where the corresponding refracted beam is elliptically polarized, see, V. C. Varghese, J. Opt. Soc. Am. 57, 90 (1967).

For experimental confirmation that the trajectory of the energy of light is the direction of the ray and not the direction of the beam, see, V. C. Varghese, J. Opt. Soc. Am. 57, 86 (1967).

Other (8)

The incident beam is always named Å, the refracted beam is named B˙, and the reflected beam is named C˙.

The subscript s indicates that the beam in the anisotropic medium is the slow beam for the given direction. The subscript attached to the associated beam, incident, reflected or refracted in an isotropic medium is w. Similarly, for the faster beam in an anisotropic medium the subscript added is ƒ and the subscript t is added to the associated beam in the isotropic medium. When the incident beam is a slow beam in an anisotropic medium and it gives rise to a slow reflected beam, the subscript added is ss and when it gives rise to a fast reflected beam the subscript added is sƒ. In the same manner, the combinations of ƒs and ƒƒ are used. The same procedure is followed for the subscripts of directions, angles, magnetic fields, and electric fields. The extra subscript r is added for those connected with a ray.

There are two different circumstances in which one of the internally reflected beams may be absent. In one case, the azimuths of the planes of polarization of the incident and reflected beams are such that all the reflected energy appears in one of the reflected beams and thus the intensity of illumination of the second reflected beam is zero. Such cases are treated in this paper. For a second case where one of the reflected beams is absent owing to "total refraction" of one component of the incident beam and where the corresponding refracted beam is elliptically polarized, see, V. C. Varghese, J. Opt. Soc. Am. 57, 90 (1967).

Positive normal to a given face and for a given incidence is the direction of the incident beam when the corresponding reflected beam retraces the path of the incident beam. Thus, the direction of the positive normal depends on the side of the surface on which light is incident. All angles of incidence, refraction, and reflection are measured in the positive counterclockwise direction from the same normal.

For experimental confirmation that the trajectory of the energy of light is the direction of the ray and not the direction of the beam, see, V. C. Varghese, J. Opt. Soc. Am. 57, 86 (1967).

In Eq. (7d), when 90°>Â>0°, both Ĉ and Ĉƒƒ, are between 90° and 180°. When 360°>Â>270°, Ĉ, and Ĉƒƒ are between 180° and 270°.

M. Planck, Theory of Light (Macmillan and Co., London, 1932), Ip. 165.

I. Todhunter and J. G. Leathem, Spherical Trigonometry Macmillan and Co., London, 1949), p. 111.

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