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.
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⌃b = tan-1 | csc (Â-B⌃) · csc (B⌃-Ĉ) · sin2B⌃ · 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.
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⌃w and α⌃t of the incident beam, (ii) γ⌃w and γ⌃t of the reflected beam, and (iii) β⌃w and β⌃t, 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.
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.
VALIAPARAMPIL C. VARGHESE, "Reflection and Refraction of Light at Internal and External Faces of Birefringent Crystals," J. Opt. Soc. Am. 57, 1351-1355 (1967)